We report the quantitative assessment of nuclear quantum effects on the strength of a single hydrogen bond formed at a water-salt interface, using tip-enhanced inelastic electron tunneling spectroscopy based on a scanning tunneling microscope. The inelastic scattering cross section was resonantly enhanced by "gating" the frontier orbitals of water via a chlorine-terminated tip, so the hydrogen-bonding strength can be determined with high accuracy from the red shift in the oxygen-hydrogen stretching frequency of water. Isotopic substitution experiments combined with quantum simulations reveal that the anharmonic quantum fluctuations of hydrogen nuclei weaken the weak hydrogen bonds and strengthen the relatively strong ones. However, this trend can be completely reversed when a hydrogen bond is strongly coupled to the polar atomic sites of the surface.