I will not say that the Quran as a whole accepts Christianity as we understand it. It actually says things like, “Exalted is He [God] above having a Son.” (Sura 4:171). That said, there are verses that support Jesus’ importance as a prophet and messiah. Mary is the only woman mentioned in the Quran – two entire suras (3 and 19) are about her.
There are obvious misunderstandings of Christian teaching (e.g., the doctrine of the trinity), which may be a result of Christian heretical sects being present around the region (e.g., tritheists against whom Jacob Barradeus sent missionaries may reflect the “say not ‘three’” in Sura 4:171 – the Arabic word used there is for “three” not, “trinity,” for example).
The correlation between many of its stories and those found in other sources available in the region from a narration-critical perspective suggests its reliance on written and oral materials that preceded the Quran (and establishment of Arabic as a written language, btw). I believe that the Quran is a pastiche assembled from various regional religious texts, including a Christian lectionary, Jewish rabbinical writings, and local stories (e.g., the story of the collapse of the Marib Dam in Yemen, which spread refugees throughout the region). Here’s an example of a narrative borrowed from the Jewish rabbinical document, the Mishnah, with the comparison between Surah 5:32 and the Mishnah Sanhedrin 4:5:
Quran: Because of that We decreed upon the Children of Israel that whoever kills a person not in retaliation or/and to spread corruption in the land
Mishnah: Therefore a single man was created in the world, to teach that if any man has caused a single soul to parish from Israel
Quran: It would be as if he killed all mankind, and if anyone saved a life, it would be as if he saved the life of all mankind
Mishnah: Scripture imputes it to him as though he had caused a whole world to perish, and if any man saves alive a single soul from Israel, scripture imputes it to him as though he had saved the whole world.
Clearly, the authors of the Quran have edited out “Children of Israel” to make the message more universal, but the textual source is pretty clear.
Its total lack of coherent organization of its suras (which are simply organized by length), and leaping from story to story within each sura suggest that the original sources have been assembled from a wide variety of contributors.