The discovery of near room temperature superconductivity in clathrate hydrides has ignited the search for both higher temperature superconductors and deeper understanding of the underlying physical phenomena. In a conventional electron-phonon mediated picture for the superconductivity for these materials, the high critical temperatures predicted and observed can be ascribed to the low mass of the protons, but this also poses nontrivial questions associated with how the proton dynamics affect the superconductivity. Using clathrate superhydride Li_{2}MgH_{16} as an example, we show through ab initio path integral simulations that proton diffusion in this system is remarkably high, with a diffusion coefficient, for example, reaching 6×10^{-6} cm^{2}/s at 300 K and 250 GPa. The diffusion is achieved primarily through proton transfer among interstitial voids within the otherwise rigid Li_{2}Mg sublattice at these conditions. The findings indicate the coexistence of proton quantum diffusion together with hydrogen-induced superconductivity, with implications for other very-high-temperature superconducting hydrides.