We predict theoretically a regime of photon-pair generation driven by the interplay of multiple bound states in the continuum resonances in nonlinear metasurfaces. This nondegenerate photon-pair generation is derived from the hyperbolic topology of the transverse phase matching and can enable orders-of-magnitude enhancement of the photon rate and spectral brightness, as compared to the degenerate regime. We show through comprehensive simulations that the entanglement of the photon pairs can be tuned by varying the pump polarization, which can underpin future advances and applications of ultracompact quantum light sources.