Black holes arise in general relativity, a classical theory of gravity.
However, we need to include quantum effects to understand black holes properly. Roger Penrose and Stephen Hawking showed thirty years ago that, according to general relativity, any object that collapses to form a black hole will go on to collapse to a singularity inside the black hole. This means that there are strong gravitational effects on arbitrarily short distance scales inside a black hole. On short disctance scales, we certainly need to use a quantum theory to describe the collapsing matter. The presence of a singularity in the classical thoery also means that once we go sufficiently far into the black hole, we can no longer predict what will happen.