By your question, I would assume you mean whether the inhabitants of Galilee believed in Jehovah, rather than participated in Judaism.

It goes very far back to the era of Joshua (about 1400 BC). Galilee was a fortified city called Kinnereth back then and was within the territory of Joshua's northern campaign. It was allocated to the tribe of Naphtali (Joshua 19:35). There are definitely some locals who converted, but the Jewish religion went through a lot of developments and wasn't finalised until much later.

The current form of modern Judaism only developed in Babylon during the exile (600 BCE) when Jews relied on religion and nationalism to remind them of their roots. Ancient Israel has always believed in Jehovah. It was just a question of whether Jehovah was their only God or one among many gods. Sure, the 10 commandments disallowed belief in other gods, but that didn't stop ancient Israel in general and its territories from cycling between monotheism and polytheism many times. They only settled on full serious monotheism during the Babylonian exile.

If your question meant how far did they go in believing in Jehovah, 1400 BCE after Joshua conquered the region. If your question meant how far did they go in believing in modern Judaism, about 530 BCE after Cyrus allowed the Jews to return to their homes, which includes Galilee.