Analogue physics became very popular in recent decades. It allows simulating inaccessible physical phenomena, such as black holes, in the laboratory. The first success of analogue physics is in fact much older being due to Maxwell, who derived his equations for the electromagnetic field by analogy with fluid dynamics in presence of vortices. Here we propose to use vortices for analogue gravity. We implement an acoustic Kerr black hole with quantized angular momentum in a Bose-Einstein condensate. We show that the condensate's metric is equivalent to the Kerr's one, exhibiting a horizon and an ergosphere. We confirm that this metric is obeyed not only by weak density waves, but also by quantum vortices which behave as massive test particles. We use these topological defects to demonstrate a quantum Penrose effect, extracting the rotation energy of the black hole by quanta of angular momentum. The particle trajectories are well described by the timelike geodesics of the Kerr metric, confirming the potential of analogue quantum gravity.