Within Christianity, the doctrine of the Trinity states that God is a single being who exists, simultaneously and eternally, as a communion of three distinct persons, the Father, the Son and the Holy Spirit.
Within Islam however, such a concept of plurality within God is a denial of monotheism, and utterly foreign to the revelation found in Muslim scripture. The act of ascribing partners to God, whether they be sons, daughters, or other partners, is considered to be blasphemous in Islam. The Qur'an repeatedly and firmly asserts God's absolute oneness, thus ruling out the possibility of another being sharing his sovereignty or nature. However, this has not prevented later Christian scholars from questioning whether the Qur'an does reject the concept of the Trinity. Yet, there has been very little doubt of this rejection by Muslims from a very early date.
Three Qur'anic verses may directly refer to this doctrine.
The Trinity in the Qur'an
The possible Qur'anic references to the doctrine of "Trinity" are verses 4:171, 5:73, and 5:116.
People of the Book, do not go to excess in your religion, and do not say anything about God except the truth: the Messiah, Jesus, son of Mary, was nothing more than a messenger of God, His word, directed to Mary, a spirit from Him. So believe in God and His messengers and do not speak of a 'Trinity' - stop, that is better for you - God is only one God, He is far above having a son, everything in the heavens and earth belongs to Him and He is the best one to trust. (Qur'an 4:171, M. A. S. Abdel-Haleem translation)
Those who say, 'God is the Messiah, son of Mary,' have defied God. The Messiah himself said, 'Children of Israel, worship God, my Lord and your Lord.' If anyone associates others with God, God will forbid him from the Garden, and Hell will be his home. No one will help such evildoers.
Those people who say that God is the third of three are defying [the truth]: there is only One God. If they persist in what they are saying, a painful punishment will afflict those of them who persist. Why do they not turn to God and ask his forgiveness, when God is most forgiving, most merciful? The Messiah, son of Mary, was only a messenger; other messengers had come and gone before him; his mother was a virtuous woman; both ate food. See how clear We make these signs for them; see how deluded they are. (Qur'an 5:72 onwards, M. A. S. Abdel-Haleem translation)
When God says, 'Jesus, son of Mary, did you say to people, "Take me and my mother as two gods alongside God"?' he will say, 'May You be exalted! I would never say what I had no right to say - if I had said such a thing You would have known it: You know all that is within me, though I do not know what is within You, You alone have full knowledge of things unseen (Qur'an 5:116, M. A. S. Abdel-Haleem translation)
Interpretation of these verses is varied and the most straightforward appears to be 5:73. It has been interpreted as a potential criticism of Syriac literature that references Jesus as "the third of three" and thus an attack on the view that Christ was divine. Hence, verses 5:72-75 may merely be criticizing the idea that Jesus and God are the same. Alternatively, it may be a purposeful simplification of the Christian belief in the humanity and divinity of Christ in order to expose its potential weakness when viewed from the firmly monotheistic position of Islam.
Similarly, verse 4:171 can be read as a rejection of the divinity of Jesus. It is worth noting that in explaining these verses, early Muslim Qur'anic commentators noted that "the Christian 'three' was an internal characteristic of the godhead... rather than a series of external beings placed together with God."
Regarding the verse 5:116, some scholars have written that the version of the "Trinity" concept that the Qur'an is criticizing appears to be God, Jesus, and Mary; and that this is not a description of orthodox Christian belief, wherein the third part of the Trinity is the Holy Spirit. Edward Hulmes writes:
"The Qur'anic interpretation of trinitarian orthodoxy as belief in the Father, the Son, and the Virgin Mary, may owe less to a misunderstanding of the New Testament itself than to a recognition of the role accorded by local Christians (see Choloridians) to Mary as mother in a special sense." 
There is also debate about whether this verse is about the Trinity. For example, Thomas states that verse 5:116 need not be seen as discussing the Christian Trinity, but rather, giving examples of shirk (claiming divinity for beings other than God) and a "warning against excessive devotion to Jesus and extravagant veneration of Mary, a reminder linked to the central theme of the Qur'an that there is only one God and he alone is to be worshipped."  When read in this light, it can be understood as an admonition, "Against the divinization of Jesus that is given elsewhere in the Qur'an and a warning against the virtual divinization of Mary in the declaration of the fifth-century
church councils that she is 'God-bearer'."