Not all supernovae have a GRB and not all GRBs have a supernova, but most long burst GRBs are thought to be from a supernova, no? I've been out of supernova research for a few years now, but that's what I recall.
So, I definitely didn't study GRBs in any depth, but based on my increasingly vague memories from my degree in astrophysics, quasars don't have much to do with it. All of the major proposed mechanisms I recall for GRBs are on the scale of a star or two (supernovae, neutron star mergers, and the like) whereas quasars are a phenomenon involving supermassive black holes in galactic nuclei. Please, definitely correct me if I'm wrong!
You're correct. It's thought that long GRBs originate in certain kinds of supernovae. (What kinds of supernovae specifically isn't really known.) Short GRBs are thought to originate in mergers between neutron stars.