Short answer
Not in the legal sense. The Quran treats Christians and Jews as People of the Book, a category separate from the polytheists (mushrikin), which is why their food and marriage to their women are lawful for Muslims. At the same time, Islam teaches that the Trinity and the worship of Jesus contradict pure monotheism, and the Quran calls on Christians to leave them.
Christians are Ahl al-Kitab in Hanafi law, even though Muslims hold that belief in the Trinity is a form of shirk in itself. The relied-upon Hanafi view keeps them in that category for food and marriage, because the Quran called them People of the Book while knowing their beliefs. Marriage to a Christian woman is valid but makruh tanzihi (disliked), and more strongly disliked in non-Muslim lands.
The four schools agree on this.
Explanation
The Quran speaks of "the People of the Scripture and the polytheists" as two different groups, and gives the People of the Book special rulings: their food is lawful for Muslims, and Muslim men may marry their chaste women. It also says they are "not all the same", praising some of them for their night prayer and devotion. The four schools agree that Christians are People of the Book.
The Quran is also clear about the belief itself. It says those who call Allah "the third of three" have disbelieved, and it criticises taking religious leaders as lords besides Allah. So Muslims believe the Trinity is a form of associating partners with God, even though Christians themselves affirm that God is one. Both truths stand side by side in the Quran: a theological disagreement, and a legal category of respect.
A few early scholars took a stricter view. Abdullah ibn Umar, in a report in al-Bukhari, would not allow marriage to a Christian woman, saying he knew of no greater shirk than a woman saying her Lord is Jesus. The majority of scholars, including the four schools, did not follow him, because the verse permitting such marriages came while the Christians already held these beliefs.
In speech, Muslims should not use "idolater" or "kafir" as an insult toward Christian neighbours. The Quran tells Muslims to argue with the People of the Book only in the best way. See who are the People of the Book? and marrying a Christian woman.
The evidence
Quran, Surah Al-Bayyina 98:1
بِسْمِ ٱللَّهِ ٱلرَّحْمَٰنِ ٱلرَّحِيمِ لَمْ يَكُنِ ٱلَّذِينَ كَفَرُوا۟ مِنْ أَهْلِ ٱلْكِتَٰبِ وَٱلْمُشْرِكِينَ مُنفَكِّينَ حَتَّىٰ تَأْتِيَهُمُ ٱلْبَيِّنَةُ ﴿١﴾
Those who disbelieved among the People of the Scripture and the polytheists were not to be parted [from misbelief] until there came to them clear evidence -
The Quran names the People of the Book and the polytheists as separate groups.
Quran, Surah Al-Maaida 5:5
ٱلْيَوْمَ أُحِلَّ لَكُمُ ٱلطَّيِّبَٰتُ ۖ وَطَعَامُ ٱلَّذِينَ أُوتُوا۟ ٱلْكِتَٰبَ حِلٌّۭ لَّكُمْ وَطَعَامُكُمْ حِلٌّۭ لَّهُمْ ۖ وَٱلْمُحْصَنَٰتُ مِنَ ٱلْمُؤْمِنَٰتِ وَٱلْمُحْصَنَٰتُ مِنَ ٱلَّذِينَ أُوتُوا۟ ٱلْكِتَٰبَ مِن قَبْلِكُمْ إِذَآ ءَاتَيْتُمُوهُنَّ أُجُورَهُنَّ مُحْصِنِينَ غَيْرَ مُسَٰفِحِينَ وَلَا مُتَّخِذِىٓ أَخْدَانٍۢ ۗ وَمَن يَكْفُرْ بِٱلْإِيمَٰنِ فَقَدْ حَبِطَ عَمَلُهُۥ وَهُوَ فِى ٱلْءَاخِرَةِ مِنَ ٱلْخَٰسِرِينَ ﴿٥﴾
This day [all] good foods have been made lawful, and the food of those who were given the Scripture is lawful for you and your food is lawful for them. And [lawful in marriage are] chaste women from among the believers and chaste women from among those who were given the Scripture before you, when you have given them their due compensation, desiring chastity, not unlawful sexual intercourse or taking [secret] lovers. And whoever denies the faith - his work has become worthless, and he, in the Hereafter, will be among the losers.
Special rulings for the People of the Book show their distinct legal status.
Quran, Surah Al-Maaida 5:73
لَّقَدْ كَفَرَ ٱلَّذِينَ قَالُوٓا۟ إِنَّ ٱللَّهَ ثَالِثُ ثَلَٰثَةٍۢ ۘ وَمَا مِنْ إِلَٰهٍ إِلَّآ إِلَٰهٌۭ وَٰحِدٌۭ ۚ وَإِن لَّمْ يَنتَهُوا۟ عَمَّا يَقُولُونَ لَيَمَسَّنَّ ٱلَّذِينَ كَفَرُوا۟ مِنْهُمْ عَذَابٌ أَلِيمٌ ﴿٧٣﴾
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.
The Quran rejects the Trinity as contrary to the oneness of God.
Quran, Surah Aal-i-Imraan 3:113
۞ لَيْسُوا۟ سَوَآءًۭ ۗ مِّنْ أَهْلِ ٱلْكِتَٰبِ أُمَّةٌۭ قَآئِمَةٌۭ يَتْلُونَ ءَايَٰتِ ٱللَّهِ ءَانَآءَ ٱلَّيْلِ وَهُمْ يَسْجُدُونَ ﴿١١٣﴾
They are not [all] the same; among the People of the Scripture is a community standing [in obedience], reciting the verses of Allah during periods of the night and prostrating [in prayer].
The Quran praises the righteous among the People of the Book.
Where the ruling is recorded
- Radd al-Muhtar (Ibn Abidin), Kitab al-Nikah, Fasl fi al-Muharramat, on marrying a woman of the Book. Discusses whether belief that Jesus is God takes a Christian out of the People of the Book.
- al-Hidaya, Kitab al-Nikah, the section on the women one may not marry. Permits marriage to a woman of the Book.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.