The accretion of material onto (into) black holes (and neutron stars) provides environments that are both very hot and (relatively) dense. Under these circumstances it is possible for nuclear fusion to occur, the question is whether this is significant, both energetically or as a means of producing new chemical elements (nucleosynthesis).
The answer to the first of these questions is relatively straightforward. As material falls towards the black hole, its angular momentum forces it to form an accretion disk. Viscous processes heat the disk and provide torques, cause the material to lose energy and angular momentum and eventually allow it to fall into the black hole. Much of the gravitational potential energy (GPE) gained as the material falls towards the black hole does end up heating the material.
The innermost stable circular orbit of a black hole is at 3 Schwarzschild radii $=6GM/c^2$, where $M$ is the black hole mass. The GPE released for material of mass $m$ falling to this radius is $\sim GMmc^2/6GM = mc^2/6$.
i.e. fully one sixth of the rest mass energy of the material could be released as heat.
Compare this with nuclear fusion. The fusion of hydrogen into helium only releases 0.7% of the rest mass as energy that can heat the accretion disk.
So from the energetic point of view, fusion reactions are negligible, unless they can occur much further out in the disk
The question about nucleosynthesis yields is more complex. The more massive a black hole and the higher the accretion rate, then in general the higher the disk temperature and density and the higher the fusion rate. But it also depends on the details of the cooling processes that are possible and how much material is advected into the black hole. Hu & Peng (2008) present some models of accretion onto a 10 solar-mass black hole and suggest that it may be possible to produce certain rare isotopes by this mechanism. Stellar-sized black holes probably need very substantially super-Eddington accretion rates to achieve the necessary temperatures to sustain nuclear fusion (i.e. much greater accretion rates than are possible by radiation-pressure opposed spherical accretion flows), according to Frankel (2016). Such rates are likely only in the cases where black holes disrupt a binary companion, rather than through a steady accretion flow.