A fiber-integrated spectrometer for single-photon pulses outside the telecommunications wavelength range based upon frequency-to-time mapping, implemented by chromatic group delay dispersion (GDD), and precise temporally-resolved single-photon counting, is presented. A chirped fiber Bragg grating provides low-loss GDD, mapping the frequency distribution of an input pulse onto the temporal envelope of the output pulse. Time-resolved detection with fast single-photon-counting modules enables monitoring of a wavelength range from 825 nm to 835 nm with nearly uniform efficiency at 55 pm resolution (24 GHz at 830 nm). To demonstrate the versatility of this technique, spectral interference of heralded single photons and the joint spectral intensity distribution of a photon-pair source are measured. This approach to single-photon-level spectral measurements provides a route to realize applications of time-frequency quantum optics at visible and near-infrared wavelengths, where multiple spectral channels must be simultaneously monitored.