Verified answer

The "flood story" and the story about "exile from Eden" were not unique to Judaism, and were part of many cultures in the ancient Middle East. The ancient Middle East was famous for its sailors, who sailed all along the Mediterranean, and even to Africa. So these stories would have reached Greece in the Biblical period, long before Greek mythology was fully developed. However, it is unlikely that Judaism itself had a significant influence on Greek mythology.

Judaism was definitely known to the Greeks by the Second Temple period, when they invaded Israel. In fact, the Jews fought the Greeks in the Hanukkah story. But Greek mythology would have been well-developed by that point.

The question of when the idea of Jewish religion reached Greece is very possibly not the correct approach to your question about the parallels in the Hebrew Bible and Greek mythology. In fact, there was no standing Jewish population in Greece until many centuries after their mythology was developed, and the very idea that the Jewish Bible had any influence whatsoever on Greek mythology is almost impossible. Though the Greeks did construct a network of trade centers around the Meditterranean, they were a very scattered people politically, and even if one city state was to come across a story from the Jewish Bible (though that itself is pretty unlikely) it would probably still have no influence on the written Greek myth and gain widespread attention. Further discouraging your theory that Greek mythology could have been influenced by the Jewish Bible is the fact that neither the Jews nor the Greeks were very expansive (that is, neither nation did much territorial conquest beyond their little world) and so their worlds remained very distant from one another until the emergence of Alexander the Great (but by the time of Alexander the Great, the Greek myths were pretty solidly set).

A more fitting approach would be to look at parallels from the Jewish Bible to the mythology of other nations. For example, the myth of Samson and the lion had its parallel in the Mesopotamian Gilgamesh as well as the Greek Hercules. And a flood story appears in Judaism as well as Greek, Mesopotamian, and Chinese myth. And these are just a few similarities. A more plausible theory than Judaism influencing Greek myth would be to look at the situation as a sort of chain: while it's unlikely Jewish influences reached Greece during the formation of their mythology, it is historically possible that Judaism may have influenced Caananite or Mesopotamian or other religions that developed in that area. And influences can be traced from Greek mythology to ancient Hittite or Babylonian tradition, which may have in turn been influenced by Judaism.

Of course, if you're religious you could bring these widespread parallels as proof that the Bible is true, and that the national myths are all based on historical events.