Quantum theory includes various relativistic quantum theories (the first and best being perhaps quantum electrodynamics). You mean general relativity, although in reality the problem is how to either formulate a theory of gravitation which can be incorporated into quantum theory that isn't general relativity, or to (as you put it), "find a compatibility) between the two. However, it doesn't matter. Black holes don't matter here: the problem is that in general relativity spacetime curvature is a "background" space of sorts within which all interactions, dynamics, physical systems, etc., exist or occur. However, in particle physics/relativistic quantum theory, all forces are mediated through particles; there is no absolute "background" space. Quantum field theories/particle physics works by "quantizing" the necessary forces and entities found in classical physics (e.g., the electromagnetic field or energy) into discrete particles, including forces. In QFT/particle physics, the quantized gravitational force (graviton particles) aren't quantizations of spacetime curvature, because spacetime curvature can't be quantized. So gravity and quantum theory exist in a very problematic relationship, as the "gravity" of quantum physics either doesn't exist or exists in a manner incompatible with the mechanism underlying gravitation in general relativity.The issue that Kaku was illustrating had to do with trying to find compatibility between Relativity and Quantum Theory.
Black holes aren't a problem of the scientific theories or mathematics involved, any more than either of these are involved in the various "impossible" technologies Kaku describes as future technologies that will be realized. That the properties of black holes, which emerge principally from the equations of general relativity, run into problems when we consider those quantum mechanical or quantum field theoretic aspects of their nature, is not a problem presented by black holes. There exists no evidence of black holes that is inconsistent with either particle physics, quantum theory, or general relativity. This is because to the extent we have experimental evidence of black holes, it is based upon the properties ascribed to them by theory that were in accordance with empirical tests. No experimental evidence from black holes present any issues with scientific theory or mathematics. The issues lie in the formulations of the underlying theories themselves, without black holes and without Kaku's issues with infinities.