The optical hybridization of localized surface plasmons and photonic modes of dielectric nanostructures provides us wide arenas of opportunities for designing tunable nanophotonics with excellent spectral selectivity, signal enhancement, and light harvesting for many optical applications. Graphene-supported Au nanoprisms on a periodic Si nanopillar array will be an ideal model system for examining such an optical hybridization effect between plasmonic modes and photonic modes. Here, through the measurement of the reflectance spectra as well as graphene phonons by surface-enhanced Raman scattering (SERS), we investigated both the far-field and near-field properties of these optically hybridized modes. The effects of photonic modes and Mie resonances of the Si nanopillars on the localized surface plasmons of the Au nanoprisms and on their near-field enhancement were experimentally elucidated through the measurements of graphene phonons using two excitation lasers with wavelengths of 532 and 785 nm. The wavelength-dependent SERS intensities of monolayer graphene are clearly understood in terms of the optical hybridization, and the SERS enhancement factor estimated from finite-difference time-domain simulations exhibited good agreement with the measurements. The elucidated spectral tunability in the near-field light-matter interaction would be useful for potential applications in various types of graphene-based photonics.