This question of Christian and Greek influence in Islamic philosophy and theology is a very touchy subject. A number of books will preface any discussion of this with the point that question itself often presumes that nothing in Islamic philosophy and theology could, itself, be original. That guys like Ibn Sina and Al-Farabi were just some sort of middle men between Aristotle, Augustus, until Aquinas and the Europeans could get it all back. Moreover the opposite question is much less asked, as Europeans tend to discount the idea that there was either an interplay and exchange of ideas, that medieval Christian theology actually gained a lot from Islam. These kinds of questions also tend to crop up with a lot of Islamic intellectual production. Islamic literature must have come from India or Persia, or be an un-Islamic outlier. As with Christianity, the idea that Europe was being influenced by Islam as Islam was being influenced by other things as part of a global exchange in the middle ages (that was of course always continuous and goes on to this day) is likewise not really discussed.

Now with all of that as preface and bearing it in mind: there were a few really interesting ways in which this was occurring. The first and perhaps most obvious was the Translation Movement of Baghdad, which is how all these Greek texts ended up in Arabic. The early days of the Translation movement were dominated by converts and native Greek speakers (edit: and Syriac speakers, as an intermediary IIRC,) but was ultimately a native Muslim endeavor and supported by the Caliph. There's a pretty straightforward exchange of ideas here.

Now these texts had a huge impact on the "Falsafa" philosophers. I think it would be a fair assessment to describe many of these thinkers (again, like Ibn Sina and Al Farabi) as being like Islamic Scholastics, trying to synthesize Islam and Greek philosophy. But the influence of these thinkers on modern Islamic theology is limited. Here (and for a whole host of reasons that would require another post altogether) al-Ghazali, who used the philosophical reasoning to combat "Falsafa," ultimately won the day.

It's worth pointing out here that much of what Europeans would call Theology, in the Islamic medieval world this was all pretty much strictly "Falsafa." The basic theological tenets were so overriding and not up for discussion that Ghazali, however influential he was in combating the philosophers, was actually removed from his teaching position for raising theological issues. These were basically not up for debate and form some of the most important maxims of Islam to this day (the most important probably being the God is truly one (the doctrine of tawhid), and all-powerful.)

These leads into another area, which is less of a transfer of knowledge like translation than what might be sort of a theological zeitgeist. Christians in this time period were very much concerned with the attributes and nature of God. The trinity was a really confusing doctrine in the middle ages (and it still is.) Muslims didn't have the problem of tri-partite God but they entered into their own, very different, but equally minute and complicated discussion of God's attributes, a lot of which revolve around statements of what "God is". e.g. if God IS All-Merciful is mercy somehow a separate attribute from him, or is Mercy itself God? How can we understand His attributes as separate from himself if he IS them? I've gone through book length discussions of these debates and they don't get any easier. It's very hard as a modern secular individual to understand how this theological minutiae (and it really does become minutiae) could be so important.

The third area I would point to that might not seem like theology has to do with imperial religious law a little bit before the time period I've discussing. Normally we probably wouldn't think of law, even religious law, as "theology" but in Islam the law plays such central role that it's worth considering. Here we have to understand the Islamic empire as a participant in a world imperial culture. Byzantine ideas, particularly in the time of the radical iconoclasts and the early schisms, made their way east and Caliphs like Harun al-Rashid and Mamun embarked on a path to codify and unify Islamic law under a single coherent imperial system. Although that struggle was unique (the mihna, under Mamun and his immediate successors, was the kind of theological inquisition that as far as I'm aware did not develop in Christianity until later) it clearly was being influenced by developments in Christendom, but in an Islamic context.

Anyways I think that's probably a good start. Let me know if you have any other questions. William Montgomery Watt's "The Formative Period in Islamic Thought" is a great primer on much of this. Benjamin Jokisch's "Islamic Imperial Law: Harun-Al-Rashid's Codification Project" covers the last bit on imperial law and the Byzantines.

I can try to recommend more but unfortunately I'm typing this from a Bolt Bus.

Edit: I'd be remiss if I didn't bring up the Asharites and the Mutazilites. Ghazali is often held as the Uber Aharite fighting off those philosophical mutazilites. That's not a bad laymen's understanding, but the Greek influences on both philosophies understandings of the nature of the physical world and therefore God's interaction with it are immense. But on these theological points the two positions were actually broadly similar, with much of the debate surrounding obscure points of Man's ability to initiate actions and to a certain extent issues over free will. They basically start from a Greek premise in a lot of these cases (as far as I understand both schools of thought were atomists) but I'm not actually familiar enough with the Greek side to know how much of this is Greek or original. I can expand on this if you want but I'd want to get off this bus and in front of some notes and old essays to make sure I'm relaying it correctly as it does really get into the weeds.