We have fabricated an L3 optical nanocavity operating at visible wavelengths that is coated with a thin-film of a fluorescent molecular-dye. The cavity was directly fabricated into a pre-etched, free-standing silicon-nitride (SiN) membrane and had a quality factor of Q = 2650. This relatively high Q-factor approaches the theoretical limit that can be expected from an L3 nanocavity using silicon nitride as a dielectric material and is achieved as a result of the solvent-free cavity-fabrication protocol that we have developed. We show that the fluorescence from a red-emitting fluorescent dye coated onto the cavity surface undergoes strong emission intensity enhancement at a series of discrete wavelengths corresponding to the cavity modes. Three dimensional finite difference time domain (FDTD) calculations are used to predict the mode structure of the cavities with excellent agreement demonstrated between theory and experiment.