Did the Quran get the Trinity wrong?
One common claim that is made against the Quran by the Christians is that the Quran doesn't understand the Trinity, that the Quran's understanding of the Trinity is wrong, and ignorant.
The Christians make the claim that Islam believed that Mary, the mother of Jesus, was a member of the Trinity. For instance they quote the following verse:
And behold! Allah will say: "O Jesus the son of Mary! Didst thou say unto men, 'Take me and my mother for two gods beside Allah'?" He will say: "Glory to Thee! Never could I say what I had no right (to say). Had I said such a thing thou wouldst indeed have known it. Thou knowest what is in my heart, though I know not what in Thine. For Thou knowest in full all that is hidden. [Qur'an 5:116]
So from this verse the Christian asserts that the Quran did not know what the Trinity was, as Mary is not in the Trinity, hence the Quran was wrong. For starters the Christians themselves have committed the mistake, not the Quran, if anyone reads Surah 5:116 (the above quoted verse) they will see that the verse says NOTHING about a Trinity. So we need to ask our Christian friends from where do they assert that this verse is mentioning the Trinity? The Christian has simply read what is not there.
In fact in the very same Surah, in passage 73, the Quran does mention the Trinity:
They do blaspheme who say: Allah is one of three in a Trinity: for there is no god except One Allah. If they desist not from their word (of blasphemy), verily a grievous penalty will befall the blasphemers among them. (5:73)
In this passage the Quran does mention the Trinity, yet it says nothing about Mary being included in the Trinity of the Christians. The Quran also mentions the Trinity in another Surah:
O People of the Book! Commit no excesses in your religion: Nor say of Allah aught but the truth. Christ Jesus the son of Mary was (no more than) a messenger of Allah, and His Word, which He bestowed on Mary, and a spirit proceeding from Him: so believe in Allah and His messengers. Say not "Trinity" : desist: it will be better for you: for Allah is one Allah: Glory be to Him: (far exalted is He) above having a son. To Him belong all things in the heavens and on earth. And enough is Allah as a Disposer of affairs. (4:171)
Here again the Quran attacks the Trinity, yet again there is nothing about Mary. So already we can see a problem with the Christian argument, the Quran when it attacks the Trinity it mentions the Trinity quite clearly, and when it does so it says nothing about Mary being involved within the Trinity. Now when the Quran mentions Jesus and Mary being worshiped as gods it never mentions the Trinity at all!
So the burden of proof is now on the Christian, show us where the Quran says anything about Mary being in the Trinity, I have quoted two passages from the Quran which attack the Trinity, and neither of these two passages say that Mary is in the Trinity.
So the Quran attacks both the Trinity, as well as Christians who have wrongly raised Mary to such a high degree that she has become like a god. Here is the proof that some Christians have turned Mary into a godly figure:
As a faithful Catholic, and later as a nun, I was devoted to Mary. The prayers and practices were so familiar. They were taught to me by sincere people. I prayed the rosary, including rosary novenas. I wore a Brown Scapular and a Miraculous Medal. (You can read about these things in the Glossary, which is Appendix C.) I visited shrines that honor Mary. I had beautiful statues of Mary. I attended special services where we prayed to Mary and recited a litany of titles honoring her. I read books about apparitions of Mary, and dreamed of visiting Lourdes and Fatima. I participated in processions honoring Mary. A statue of Mary was put on a platform that was decorated with flowers. There were poles on the platform, so that men could carry it. The men walked through the streets, carrying the statue on the platform. We walked behind the statue, singing songs in Mary's honor.
When Pope John Paul II was shot, while the ambulance was rushing him to the hospital, the Pope was not praying to God or calling on the name of Jesus. He kept saying, over and over: "Mary, my mother!" Polish pilgrims placed a picture of Our Lady of Czestochowa on the throne where the Pope normally sat. People gathered around the picture. Vatican loudspeakers broadcasted the prayers of the rosary. When the Pope recovered, he gave Mary all the glory for saving his life, and he made a pilgrimage to Fatima to publicly thank her. (Note 4)
One popular prayer in Mary's honor is the "Hail Holy Queen," which is known in Latin as the "Salve Regina". It is traditionally included as part of praying the rosary.
"Hail, holy Queen, Mother of Mercy! Our life, our sweetness and our hope! To thee do we cry, poor banished children of Eve. To thee do we send up our sighs, mourning and weeping, in this valley of tears. Turn, then, most gracious Advocate, thine eyes of mercy toward us; and after this our exile show unto us the blessed fruit of thy womb, Jesus. O clement, O loving, O sweet Virgin Mary."
Alfonsus de Liguori (1696-1787) was a principal proponent of the Marianist Movement, which glorifies Mary. He wrote a book entitled "The Glories of Mary" which is famous, influential and widely read. In this book, de Liguori says that Mary was given rulership over one half of the kingdom of God; Mary rules over the kingdom of mercy and Jesus rules over the kingdom of justice. De Liguori said that people should pray to Mary as a mediator and look to her as an object of trust for answered prayer. The book even says that there is no salvation outside of Mary. Some people suggest that these views are extreme and not representative of Catholic Church teaching. However, instead of silencing de Liguori as a heretic, the Catholic Church canonized him as a saint and declared him to be a "doctor of the Church" (a person whose teachings carry weight and authority). Furthermore, his book is openly and officially promoted by the Catholic Church, and his teachings have influenced popes. (Note 9)
Pope Benedict XV said of Mary that "[O]ne can justly say that with Christ, she herself redeemed mankind." (Note 10) Pope Pius IX said: "Our salvation is based upon the holy Virgin... so that if there is any hope and spiritual healing for us we receive it solely and uniquely from her." (Note 11)
Also from http://www.newadvent.org/cathen/15464b.htm
Mary's Divine motherhood
Mary's Divine motherhood is based on the teaching of the Gospels, on the writings of the Fathers, and on the express definition of the Church. St. Matthew (1:25) testifies that Mary "brought forth her first-born son" and that He was called Jesus. According to St. John (1:15) Jesus is the Word made flesh, the Word Who assumed human nature in the womb of Mary. As Mary was truly the mother of Jesus, and as Jesus was truly God from the first moment of His conception, Mary is truly the mother of God. Even the earliest Fathers did not hesitate to draw this conclusion as may be seen in the writings of St. Ignatius , St. Irenaeus , and Tertullian . The contention of Nestorius denying to Mary the title "Mother of God"  was followed by the teaching of the Council of Ephesus proclaiming Mary to be Theotokos in the true sense of the word. 
For the attitude of the Churches of Asia Minor and of Lyons we may appeal to the words of St. Irenaeus, a pupil of St. John's disciple Polycarp ; he calls Mary our most eminent advocate. St. Ignatius of Antioch, part of whose life reached back into apostolic times, wrote to the Ephesians (c. 18-19) in such a way as to connect the mysteries of Our Lord's life more closely with those of the Virgin Mary. For instance, the virginity of Mary, and her childbirth, are enumerated with Christ's death, as forming three mysteries unknown to the devil. The sub-apostolic author of the Epistle to Diognetus, writing to a pagan inquirer concerning the Christian mysteries, describes Mary as the great antithesis of Eve, and this idea of Our Lady occurs repeatedly in other writers even before the Council of Ephesus. We have repeatedly appealed to the words of St. Justin and Tertullian, both of whom wrote before the end of the second century.
As it is admitted that the praises of Mary grow with the growth of the Christian community, we may conclude in brief that the veneration of and devotion to Mary began even in the time of the Apostles.
So it is quite clear that many Christians did treat Mary as a divine godly being, hence the Quran condemned this practice, as well as condemning the Trinity, the Quran condemns both.
Addressing an Objection
Now some may say that if one reads the context of Surah 5:73, when the Quran attacks the Trinity, one will see that the context shows that the author of the Quran believed that Mary was part of the Trinity, we quote the context:
They do blaspheme who say: Allah is one of three in a Trinity: for there is no god except One Allah. If they desist not from their word (of blasphemy), verily a grievous penalty will befall the blasphemers among them. Why turn they not to Allah, and seek His forgiveness? For Allah is Oft-forgiving, Most Merciful. Christ the son of Mary was no more than a messenger; many were the messengers that passed away before him. His mother was a woman of truth. They had both to eat their (daily) food. See how Allah doth make His signs clear to them; yet see in what ways they are deluded away from the truth! (5:73-75)
So some Christians may assert from verse 76, where Mary is mentioned, that the Quran does believe that Mary was part of the Trinity. They will assert that since the Quran makes the point that both Mary and Jesus ate food, the Quran here is trying to disprove divinity from both Jesus and Mary, and with the Trinity being mentioned in verse 73, this must mean the author thought that Mary was in the Trinity.
As we saw above there are Christians who believe in Mary's ?Divine Motherhood' that she was the mother of God. Hence the Quran here is trying to show the fallacy in such a belief, how can there be a Trinity, and Mary having a divine motherhood due to giving birth to a supposedly divine Jesus, when Mary was a normal lady like every other lady, eating food and so on. In case Christians don't get it, Jesus' divinity and the Trinity itself is what makes Mary so special and given a divine status, yet the Quran is arguing how such a thing possible in the first place, Mary was like other humans, she ate food and so on, so a lady like herself with never give birth to God in the first place, the very same God whom Christians claim to be a part of the Trinity.
The verse is also going on to show the complete humanity of Jesus which refutes the Trinity, by emphasizing the humanity of Mary, the Quran is trying to emphasize the humanity of Jesus as well, his mother was a normal human being like everyone else, hence why would she give birth to the almighty God and part of what is called the Trinity? It doesn't add up at all, and this is precisely what The Quran is getting at, Mary was a normal lady, hence she doesn't give birth to God or anything to do with divinity!
So the verse is not including Mary in the Trinity, rather it is attacking the very core and basis of the Trinity by pointing out that it is a complete fallacy for a normal lady who eats, to give birth to God himself, the very same God who supposedly makes up a Trinity.
So in conclusion Mary is never mentioned in the Trinity, when the Quran mentions the worship of Jesus and Mary, that same verse says nothing about a Trinity. The Quran does mention the Trinity in two other places, and Mary is never included within the Trinity itself, hence the Quran is not wrong, it is the Christian who is wrong, and it is the Christian who has mis-understood the Quran.
And Allah Knows Best!