We develop a general theoretical framework of integrated paired photon-plasmon generation through spontaneous wave mixing in nonlinear plasmonic and metamaterial nanostructures, rigorously accounting for material dispersion and losses in the quantum regime through the electromagnetic Green function. We identify photon-plasmon correlations in layered metal-dielectric structures with 70% internal heralding quantum efficiency and reveal a novel mechanism of broadband generation enhancement due to topological transition in hyperbolic metamaterials.