We present a general framework that computes the two-photon spontaneous emission rate of a quantum emitter close to an arbitrary photonic structure beyond the dipolar approximation. This is relevant for strongly confined light fields, such as in plasmonic nano- and picocavities, which are currently being explored to enhance higher-order light-matter interactions. In our framework, the emitter contribution to this process is calculated analytically, while the influence of the photonic environment is determined via the computation of Purcell factors with conventional electromagnetic simulations, which avoids tedious analytic calculations for the environment. Also, our framework efficiently handles asymmetric structures that were not treated before. We show that placing a hydrogen-like emitter close to a silver nanodisk enhances the transition rate between two spherically symmetric states by 5 and 11 orders of magnitude via electric dipole and quadrupole two-photon transitions, respectively. In the future, controlling this process promises efficient entangled two-photon sources for quantum applications, new platforms in spectroscopy, as well as broadband absorbers and emitters.