Can nuclear fusion happen near a black hole?

Not very easily.

To produce thermonuclear reactions at a rate similar to the sun's, you need temperatures in excess of about 1 million degrees to produce deuterium fusion..the lowest energy fusion reaction we know about. To get 1 million degrees, particles have to be traveling at speeds of about 100 kilometers/sec since 1/2 mV^2 = kT where we have used the mass of a deuterium nucleus (m), and Boltzman's constant (1.38 x 10^-16). So, per particle of deuterium, this is about what their random speeds have to be.

For black holes, you only would get fusion if this were the random speed of particles within their own rest frames as they were falling through the horizon. Such random speeds would not be the case near supermassive black holes because their horizons are so enormous and their tidal forces are so weak. But near stellar mass black holes, gravitational tidal forces are substantially higher and it might be possible for some of the gas to reach these kinds of conditions, at least in a limited volume of space near the horizon. A signature of this would be a black hole emitting gamma rays. Black holes such as these are powerful emitters of X-rays, and some produce even harder radiation, but I know of no black holes that emit significant amounts of gamma rays.