In addition to its 2D form, the 0D form of graphene, which is also called graphene nano-flake (GNF) or nano-dot, also exists in a stable form but has been studied much less extensively than its 2D counterpart. We propose a rigorous full quantum-mechanical approach that is beyond the conventional nearest-neighbor coupling approximation, to study for the first time the linear and nonlinear optical properties of GNFs. As compared to commonly used approaches, which are based on solving Maxwell equations and are only valid for GNFs with size larger than a few tens of nanometers, our method goes significantly beyond the state-of-the-art as we provide a full quantum mechanical description of both linear and nonlinear properties of nanometer-sized GNFs. We find that the optical response of GNFs, namely the linear and nonlinear polarizabilities of GNFs, depends crucially on their shape, size, and symmetry properties. Specifically, increasing the size of nanoflakes is found to shift their accommodated quantum plasmon oscillations to lower frequency. Importantly, we show that by embedding a cavity into GNFs, one can change their symmetry properties, tune their optical properties, or enable otherwise forbidden second-harmonic generation processes.