Are you asking about a specific date or just a general "In this century..."
Also the term Byzantium... might not be the best. "Byzantium" is the name given to the Eastern Roman Empire and largely used to describe the period in which it was the the successor state to Rome (when it was predominantly Greek and the only serious claimant was the not-very-holy, not-very-roman, and not-really-an-empire Holy Roman Empire) and wasn't a term used by the Byzantines themselves.
Anyways, Christianity spread perhaps fastest among the Greeks, to the point that Constantine founded Constantinople (or Istanbul, or "New Rome", whatever suits your fancy) in part because the Roman Empire was almost entirely Christianized in its Greek Provinces while still predominantly Pagan in the west.
Saint Paul can perhaps be most credited with bringing Christianity "to the gentiles", beginning to preach in Syria between 39-43 A.D. and entered Antioch, a predominantly Greek city, in 44 A.D. In 48 A.D. he went to Iconium where he began to preach.
It's important to note that while he preached predominantly in synagogues, the Bible makes mention of Greeks attending as well, specifically in Iconium.
Acts 14:1-7 In Iconium they entered the Jewish synagogue together and spoke in such a way that a great number of both Jews and Greeks came to believe, although the disbelieving Jews stirred up and poisoned the minds of the Gentiles against the brothers. So they stayed for a considerable period, speaking out boldly for the Lord, who confirmed the word about his grace by granting signs and wonders to occur through their hands. The people of the city were divided: some were with the Jews; others, with the apostles. When there was an attempt by both the Gentiles and the Jews together with their leaders, to attack and stone them, they realized it and fled to the Lycaonian cities of Lystra and Derbe and to the surrounding countryside, where they continued to proclaim the good news.
Of note is that it places the blame mostlyon "The Jews" rather than the Greek Gentiles, this might in part be because to the Jews this was a "heresy" while to the Greeks it was a brand new religion, and that for the most part Paul had converted Gentiles rather than solely Jews.