Structural and optical properties of films and particles prepared from partially fluorinated graphene suspensions were examined. Photoluminescence (PL) coming from partially fluorinated graphene suspensions (quantum dots with fluorinated edges) and films prepared from such suspensions was observed. The necessary conditions for excitation of photoluminescence and PL emission spectra consisting of one to three PL features with energies 2.65, 2.81, and 2.97 eV have been identified. The PL emission spectra were compared with the spectrum of the size quantization levels in partially fluorinated graphene and few-layer graphene films. The spectrum of levels was revealed from the charge deep-level transient spectroscopy measurements taken during studying the electron capture and the emission processes in such films. The PL emission spectra were interpreted with the assumption that fluorination of graphene suspensions leads to formation of quantum dots within graphene. A qualitative model was proposed to explain the observed correlation between the PL emission spectrum and the energy spectrum of graphene quantum dots embedded in a fluorinated graphene matrix.