The Bible says Tyre and Sidon were surrounded by the Tribe of Asher, but not part of them. 1 Kings 11 says that a woman from Sidon Solomon married helped lead him into idolatry. Queen Jezebel was also from Sidon. The Bible says a lot of negative things about Tyre. Otherwise the Phoenicians are very close cousins to the Hebrews, speaking a very similar language (Phoenician/Punic is the closest related language to Biblical Hebrew) and sharing a similar culture outside of the whole Judaism thing (they worshipped the same gods in similar ways outside of the whole henotheism of the Jews). If the Israelites conquered them, then they might be incorporated into the Israelite nation and thus worship Yahweh as well, but you'd be permanently changing the foundations of Judaism assuming there even is a "Judaism" to speak of.
But could they? Solomon traded pretty heavily with them to build the First Temple, and clearly both sides benefitted greatly since it helped Tyre become a major city. And then you have the fact that the united Kingdom of Israel was probably very, very loosely organised, and thus the Phoenicians would be among the first to assert independence once Solomon/whoever died.