It's a bit worn, but that is the patch of VPB-102, a U.S. Navy patrol bomber unit that was in existence from 1943 to 1994. The patch depicts a green dragon hurling a bomb while perching on top of a cloud (VPB-102 was first known as the "Dragons," and then as the "Blue Geese"). During WWII, VPB-102 flew PB4Y-1s (the Navy designation for the stock B-24 Liberator) on missions from Hawaii and islands in the central Pacific. A rather detailed history of the squadron can be found here, from pages 133 to 142. The PDF is very large, so it might take a while to open.

Source:

• Roberts, Michael D. (2000). Dictionary of American Naval Aviation Squadrons, Volume 2, Chapter 3, Section 5: Patrol Squadron Histories for 3rd VP-22 to 3rd VP-25. Washington, D.C.: Naval Historical Center, Department of the Navy. pp. 133–142.