Praise be to Allah.
Yes, this belief is mentioned in the Quran, but that is by way of refutation and proving it to be false, and describing the one who believes in it as a kaafir and mushrik. Allah says (interpretation of the meaning):
“Surely, in disbelief are they who say that Allah is the Messiah, son of Maryam (Mary)” [Al-Maa'idah 5:17]
“Surely, disbelievers are those who said: ‘Allah is the third of the three (in a Trinity).’ But there is no Ilaah (god) (none who has the right to be worshipped) but One Ilaah (God —Allah). And if they cease not from what they say, verily, a painful torment will befall on the disbelievers among them” [Al-Maa'idah 5:73]
“And the Jews say: ‘Uzair (Ezra) is the son of Allah, and the Christians say: Messiah is the son of Allah. That is their saying with their mouths, resembling the saying of those who disbelieved aforetime. Allah’s Curse be on them, how they are deluded away from the truth!
They (Jews and Christians) took their rabbis and their monks to be their lords besides Allah (by obeying them in things which they made lawful or unlawful according to their own desires without being ordered by Allah), and (they also took as their Lord) Messiah, son of Maryam (Mary), while they (Jews and Christians) were commanded [in the Tawraat (Torah) and the Injeel (Gospel)] to worship none but One Ilaah (God — Allah) Laa ilaaha illa Huwa (none has the right to be worshipped but He). Praise and glory be to Him (far above is He) from having the partners they associate (with Him)” [Al-Tawbah 9:31]
This is something that is well known among the Muslims, and they are unanimously agreed that the Christians are kaafirs, and even that those who do not regard them as kaafirs are also kaafirs. Shaykh Muhammad ibn ‘Abd al-Wahhaab said concerning the things which are unanimously agreed to nullify Islam:
“Whoever does not regard the mushrikoon as kaafirs, or doubts that they are kaafirs, or thinks their religion is correct, is himself a kaafir.” (See also question no. 31807).
We are surprised at the question, in which the questioner appears to think that the shirk which exists among the Christians also exists in the religion of the Muslims. Hence we advise the questioner to read the books of ‘aqeedah which explain Tawheed (the Oneness of Allah) and discuss its meaning and rulings, and also describe the various kinds of shirk. He should also listen to useful tapes on this topic, for this is one of the most important obligations that are enjoined upon people. This trinity in which the Christians believe is not one of the things that lead to shirk, rather it is the essence of shirk. The trinity that the later Christians have fabricated has no evidence in reason and fitrah (the innate nature of man), and there is nothing to support it in the divinely-revealed Books which Allah sent down.
Ibn al-Qayyim said:
This is the attitude all the people of misguidance with their leaders and followers. When a monotheist (believer in Tawheed) disputes with the ignorant Christians and points out that their trinity is false and self-contradictory, they say, “The answer is with the priest.” The priest says, “The answer is with the archbishop.” The archbishop says, “The answer is with the patriarch.” The patriarch says that the answer is with the three hundred and eighteen clergymen who gathered at the time of Constantine and enjoined upon the Christians this trinity and shirk which contradicts sound reasoning and religion… "(Miftaah Daar al-Sa’aadah, 2/148).
With regard to the word “trinity” (tathleeth) it is not mentioned in either the Quran or the Sunnah, rather the word tathleeth (in the sense of doing something three times) is mentioned by the scholars when they discuss cleaning onself with pebbles after relieving oneself, or when discussing wudoo’, ghusl, washing the dead, saying tasbeeh when bowing and prostrating, asking permission to enter a house, etc.
What is meant in all of these cases is doing something three times, and it has nothing to do with the trinity of the Christians, which Allah mentions describing their words and commanding them to believe that He is One (Tawheed) and to believe that ‘Eesa (Jesus) is a Messenger and not a god.
Shaykh al-Islam Ibn Taymiyah said:
Allah says (interpretation of the meaning):
“O people of the Scripture (Christians)! Do not exceed the limits in your religion, nor say of Allah aught but the truth. The Messiah ‘Eesa (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 (Rooh) created by Him; so believe in Allah and His Messengers. Say not: ‘Three (trinity)!’ Cease! (it is) better for you” [Al-Nisa’ 4:171]
In this verse Allah referred to ‘trinity and incarnation, and forbade both of these beliefs. He states that the Messiah is only “a Messenger of Allah and His Word, (‘Be’ — and he was) which He bestowed on Maryam (Mary) and a spirit (Rooh) created by Him”. And He says, “so believe in Allah and His Messengers”, then He says, “Say not: ‘Three (trinity)!’ Cease! (it is) better for you”."(Al-Jawaab al-Saheeh, 2/15)
Some Christians – because of their ignorance – think that the plural pronoun which is used to show respect in verses such as “Verily, We have given you (O Muhammad) a manifest victory” [Al-Fath 48:1] and “Verily, We have sent it down as an Arabic Quran…” [Yoosuf 12:2] is proof of their false belief in trinity.
Shaykh al-Islam Ibn Taymiyah said:
The view of the salaf (early generations) of this ummah and of its imams and later generations is that the Prophet (peace and blessings of Allah be upon him) heard the Quran from Jibreel, and Jibreel heard it from Allah. The use of plural forms in such phrases is the style of Arabic speech used to refer to one who is of high standing and has helpers who obey him. So if his helpers do something by his command, he says, “we did it”. This is like when a king says, “We conquered this land, we defeated this army” and so on. Because he did that through the actions of his helpers. Allah is the Lord of the angels and they speak not until He has spoken, and they act in accordance with His commands; they do not disobey the commands of Allah, rather they do what He commands. Moreover He is their Creator and the creator of their deeds and their power. But He has no need of them; He is not like a king whose helpers do things by their own strength. So what He says when He does something through His angels is, “We did it”, this is more appropriate and He is more entitled to say it than some king.
This plural form is ambiguous and the Christians try to use it as evidence against the Prophet (peace and blessings of Allah be upon him), when they find such phrases in the Quran as “Verily, We have given you (O Muhammad) a manifest victory” [al-Fath 48:1], etc. But Allah condemned them for ignoring the clear verses in the Quran which state that God is One, but they cling to the ambiguous verses which may be interpreted as referring to one who has a peer with him, or to one who has helpers who are his slaves and creation. They follow the ambiguous verses, seeking to stir up confusion in this manner. This is confusion in the heart, by thinking that there are many gods, and seeking to twist the meaning. No one knows the true interpretation except Allah and those who are well versed in knowledge."( Majmoo’ al-Fatawa, 5/233, 234)
See also question no. 606
And Allah knows best.