The Magi may also have heard of Jewish prophecy concerning the Messiah as there were Jews living, as Acts 2:5 says, in every nation under heaven.
Many Biblical scholars believe the influence went in the other direction, from the Persians to the Jews. During the Babylonians captivity, the Babylonians were conquered by the Persians. The Persian king Cyrus let all the tribes captured by the Babylonians return to their homelands, including the Jews.
Whichever way the influence may have gone, the Magi headed the star and came to Bethlehem to worship the Christ child.