Nonlinear optical processes associated with even-order nonlinear susceptibilities are critical for both classical and quantum technologies. Inversion symmetry, however, prevents nonlinear optical responses mediated by even-order susceptibilities in several material systems pertinent for applications in nanophotonics. Here, we demonstrate induced nonlinear optical processes, namely second- and fourth-harmonic generation that are naturally forbidden in an inversion symmetric system, by strongly coupling to a photon mode of a high-Q optical cavity. As an illustrative system with an inversion symmetry, we consider a semiconductor quantum ring of GaAs that features a single effective electron. For the coupled system, we control the inversion symmetry breaking by changing the light-matter coupling strength which at the same time allows to tune the nonlinear conversion efficiency. We find that the harmonic generation yield can be significantly increased by increasing the light-matter coupling strength in an experimentally feasible way. In the few-photon limit where the incident pump field is a coherent state with just a few photons, we find that the harmonic conversion efficiency is increased for strong coupling as opposed to using intense pump fields. This new approach is applicable to a wide variety of centrosymmetric systems as the symmetry breaking rest on the properties of the photonic environment used to achieve strong light-matter interaction. Our work constitutes a step forward in the direction of realizing physically forbidden nonlinear optical processes in centrosymmetric materials widely adopted for applications in integrated photonics.