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My (flawed) understanding is the innermost rings of the accretion disk spin around the event horizon at nearly the speed of light, and it is friction (that is material collisions) that raise temperatures so high. So when a bunch of mass (say, a rogue asteroid) falls towards a black hole, it gets broken down and spaghettified by tidal forces and then combines into the accretion disk. As it moves towards the center, the friction creates a lot of light and heat which we can detect, and have called a quasar (a quasi-stellar object).
So it sure seems to fit all the parameters that make fusion likely. I can't say if we've ever detected fusion within a quasar event.
Is that really why they're called Quasars? I had no idea! TIL :-)