Considering that Trinitarianism kind of violates the idea of not associating God with anything/anyone
2 Answers
Yes, Christians can become dhimmis if they wish, regardless of their sect. Trinitarians are still considered 'People of the Book' despite their beliefs, see: Are Christians who believe in the Trinity considered People of the Book?
يا أهل الكتاب لا تغلوا في دينكم ولا تقولوا على الله إلا الحق إنما المسيح عيسى ابن مريم رسول الله وكلمته ألقاها إلى مريم وروح منه فآمنوا بالله ورسله ولا تقولوا ثلاثة انتهوا خيرا لكم إنما الله إله واحد سبحانه أن يكون له ولد له ما في السماوات وما في الأرض وكفى بالله وكيلا
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.
Being 'People of the Book' does not mean that they do not associate others with Allah, rather all non-muslims associate others with Allah in some way or another. 'People of the Book' simply means those who claim to follow the Torah and Gospel.
And Jizya and hence the pact of Dhimma is acceptable from all the disbelieving 'People of the Book' according to the Quran:
قاتلوا الذين لا يؤمنون بالله ولا باليوم الآخر ولا يحرمون ما حرم الله ورسوله ولا يدينون دين الحق من الذين أوتوا الكتاب حتى يعطوا الجزية عن يد وهم صاغرون
Fight those who do not believe in Allah or in the Last Day and who do not consider unlawful what Allah and His Messenger have made unlawful and who do not adopt the religion of truth from those who were given the Scripture - [fight] until they give the jizyah willingly while they are humbled.
The Prophet ﷺ made the pact of Dhimma with the Christians of Najran after the decided to not do mubahala on the divinity of Jesus عليه السلام and agreed to pay Jizya. And he also made a pact with the people of Yemen and Bahrain etc. and the Sahaba accepted Jizya and made the pact of Dhimma for example with the Christians of Jerusalem and other areas of Iraq and Syria, despite the fact that these people believed in the divinity of Jesus عليه السلام.
Ref:
فأهل الكتاب، اليهود والنصارى ... وفرق النصارى؛ من اليعقوبية ، والنسطورية، والملكية ، والفرنج والروم والأرمن وغيرهم، ممن دان بالإنجيل، وانتسب إلى دين عيسى والعمل بشريعته، فكلهم من أهل الإنجيل
The People of the Book are the Jews and Christians ... and the Christians include the Jacobites, Nestorians, Melkites, Franks, Romans and the Armenians and others whoever takes his religion from the Gospel and associates himself with Jesus and follows his shariah they are all the people of the Gospel
— Mughni
Trinity existed in Christianity during the time of the prophet.
Quran 5:73 -
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.
I don't believe the Trinity is seen as anything distinct in Islamic Christian laws because it has been central to Christianity since the time of Jesus.
