I have read that Islam is antitrinitarian as several verses of the Qur'an teach that the doctrine of trinity is blasphemous. What parts in the Qur'an specifically or directly discuss the trinity?
The Qur'an refers explicitly to some concept of a trinity at least twice:
They have certainly disbelieved who say, " Allah is the third of three." And there is no god except one God. And if they do not desist from what they are saying, there will surely afflict the disbelievers among them a painful punishment. Al-Ma'idah:73
O People of the Scripture, do not commit excess in your religion or say about Allah except the truth. The Messiah, Jesus, the son of Mary, was but a messenger of Allah and His word which He directed to Mary and a soul [created at a command] from Him. So believe in Allah and His messengers. And do not say, "Three"; desist - it is better for you. Indeed, Allah is but one God. Exalted is He above having a son. To Him belongs whatever is in the heavens and whatever is on the earth. And sufficient is Allah as Disposer of affairs. An-Nisa':171
At several other locations the Qur'an refers to people taking Jesus (pbuh) as a deity:
And [beware the Day] when Allah will say, "O Jesus, Son of Mary, did you say to the people, 'Take me and my mother as deities besides Allah ?'" He will say, "Exalted are You! It was not for me to say that to which I have no right. If I had said it, You would have known it. You know what is within myself, and I do not know what is within Yourself. Indeed, it is You who is Knower of the unseen. Al-Ma'idah:116
Of course in this case it doesn't refer to the traditional trinity.
Adding to other answers, although it doesn't discuss specifically the concept of trinity, Surat Al-'Ikhlāş denies trinity by the means of it's implications. Allah (c.c) says:
This ayah is commonly translated as
Say, "He is Allah , [who is] One,. Although the word
Ahad translated as "one", the word implies something that has no relatives, no similiars, no opposites, something that is absolutely unique in its kind. By its implications, Allah's "ahad"ness denies the concept of trinity.
Moreover, Allah (c.c) says;
In this ayah, Allah explains more about its "ahad"ness by saying
He neither begets nor is born, (Sahih International translation of above ayah)
Therefore, we can conclude by Surat Al-'Ikhlāş that Islam's understanding of tawheed doesn't allow the concept of trinity.
This is not so much 'discussed' in great detail, as warned of in the following verses:
O people of the Scripture (Christians)! Do not exceed the limits in your religion, nor say of Allah aught but the truth. The Messiah 'Îsa (Jesus), son of Maryam (Mary), was (no more than) a Messenger of Allah and His Word, ("Be!" - and he was) which He bestowed on Maryam (Mary) and a spirit (Ruh ) created by Him; so believe in Allah and His Messengers. Say not: "three (trinity)!" Cease! (it is) better for you. For Allah is (the only) One Ilah (God), glory be to Him (Far Exalted is He) above having a son. To Him belongs all that is in the heavens and all that is in the earth. And Allah is All-Sufficient as a Disposer of affairs.
An-Nisa, Chapter #4, Verse #171
Surely, disbelievers are those who said: "Allah is the third of the three (in a Trinity)." But there is no Ilah (god) (none who has the right to be worshipped) but One Ilah (God -Allah). And if they cease not from what they say, verily, a painful torment will befall on the disbelievers among them.
Al-Maeda, Chapter #5, Verse #73
I have provided links to the Surah's so you can see the verses in context, so as to avoid any confusion insha Allah.