i would assume they have the same cooler. Some core 2 duo's have crappy coolers though. Does the cooler have fins that kind of spiral outwards and a copper core? If so then it sounds like it should be ok. You should also remove the old thermal paste and apply new paste or it will get hot.
The Q6600's cooler was heavier than the cooler for an E6600 or E6850 which would indicate a larger copper core so using the lightweight E6300's cooler might result in overheating as the B3's ran a little hotter than the G0's.