I have been arguing with some secularist non believers and basically what they try to say is that the Quran could have been formulated because the Prophet Muhammad may have come into contact with Greek science and Greek Philosophy. Now, from what I know, by the 300s A.D. there were 2 schools in Mesopotamia or mordern Iraq, which taught the teachings of Aristotle, Ptolemy, and a couple others, mostly Aristotle though. Syria also was supposedly under Greek rule since the time of Alexander and there was Greek culture and influence there, people began speaking Greek. By the late 200s though there was a schism and rejection of Greek culture. Now, there are some things that the Greeks spoke of that can sometimes appear similar to statements in the Quran, regarding the orbits of stars and planets, the rising of the sun and how it occurs, sun and moon, embryology. From what I have seen though, the Quran makes consistent and totally accurate statements without incorporating a single one of the very many flawed theories and ideas people had back then. 

Plus there are other scientific statements which people had not yet found out at all. Can you please tell me though, in general, how much could Muhammad have learned about Greek science in his time and place? What was the magnitude of Greek influence in Northern and Southern Arabia?

There have been waves of Greek influence extending to the various countries now considered as Arab, but a proper encounter between Muslim and Greek thought only took place during the Abbasid period, i.e. one and a half centuries after the Hijrah. There has been a Jewish and Christian presence in the Hijaz (Arabian peninsula) during the time of the Prophet Muhammad - peace be upon him, but there is no evidence to suggest any meaningful Hellenistic influence. Many people have tried to rewrite history faced with the rapid success of Islam within a generation after its first proclamation. If the ideas contained in the Qur'an were merely reiterations of already existing philosophies, the likelihood of having had this tremendous impact upon the local populations is slim. The dynamic effect of the Qur'an is precisely due to the fact that it was so very different from what people were already accustomed to and that it challenged the old order so profoundly.