There is little known about the Girgashites. They are supposed to have dwelt near the Sea of Chinneroth (Galilee). The Girgashites descended from Canaan, the fourth son of Ham, the son of Noah. The name of Canaan's fourth son was Gergesus, according to Josephus. Canaan was cursed by Noah, and this curse passed onto his sons. Moses gave the command that the Girgashites were to be utterly destroyed.
Deuteronomy 7:1-2 - When the LORD your God brings you into the land where you go to possess it, and casts out many nations before you, the Hittite, the Girgashite, the Amorite, the Canaanite, the Perizzite, the Hivite, and the Jebusite, seven nations greater and mightier than you; and when the LORD your God delivers them up before you, and you strike them; then you shall utterly destroy them. You shall make no covenant with them, nor show mercy to them;
The Girgashites were one of the seven nations that God commanded to utterly destroy. They were indeed driven out by the conquering Israelites, and not much is known about them.
The religion of the ancient Canaanites was idolatrous. They made their sons and their daughters to pass through the fire to Moloch, and they worshipped Baal and Ashtaroth, which included temple prostitution.
Deuteronomy 7:1 - When the LORD thy God shall bring thee into the land whither thou goest to possess it, and hath cast out many nations before thee, the Hittites, and the Girgashites, and the Amorites, and the Canaanites, and the Perizzites, and the Hivites, and the Jebusites, seven nations greater and mightier than thou;
Nehemiah 9:8 - And foundest his heart faithful before thee, and madest a covenant with him to give the land of the Canaanites, the Hittites, the Amorites, and the Perizzites, and the Jebusites, and the Girgashites, to give [it, I say], to his seed, and hast performed thy words; for thou [art] righteous:
Joshua 24:11 - And ye went over Jordan, and came unto Jericho: and the men of Jericho fought against you, the Amorites, and the Perizzites, and the Canaanites, and the Hittites, and the Girgashites, the Hivites, and the Jebusites; and I delivered them into your hand.
Joshua 3:10 - And Joshua said, Hereby ye shall know that the living God [is] among you, and [that] he will without fail drive out from before you the Canaanites, and the Hittites, and the Hivites, and the Perizzites, and the Girgashites, and the Amorites, and the Jebusites.
Genesis 15:21 - And the Amorites, and the Canaanites, and the Girgashites, and the Jebusites.