The Gemmorah in Jerusalem Rosh Hashana 6a explains the process of adopting the Babylonian names on Months instead of Biblical ones.
דא"ר חנינה שמות חדשים עלו בידם מבבל
R'Hanina said: the names of the Months were brought with them from Babylon
The WIKI on Babylonian_calendar states that although every month is associated with a deity, Tammuz specifically bears the name of their god. The WIKI also says that the name of the god was widely known (and even worshipped) by the Israelites:
The cult of Ishtar and Tammuz may have been introduced to the Kingdom of Judah during the reign of King Manasseh and the Old Testament contains numerous allusions to them. Ezekiel 8:14 mentions Tammuz by name: "Then he brought me to the door of the gate of the Lord's house which was toward the north; and, behold, there sat women weeping for Tammuz. ..."
Tammuz seems to be the name of a deity or connected to idolatry. How is it that it became the name of a Jewish month given the prohibition of וְשֵׁ֨ם אֱלֹהִ֤ים אֲחֵרִים֙ לֹ֣א תַזְכִּ֔ירוּ לֹ֥א יִשָּׁמַ֖ע עַל־פִּֽיךָ, namely, that it is forbidden to speak the names of such deities?