First-principle molecular dynamics simulations of methane hydrate MH-III, including the quantum properties of the hydrogen nuclei, were carried out at pressures in the 5-65 GPa range, in order to observe the H-bond symmetrization at high pressure and at room temperature. According to our simulations, the symmetrization transition takes place around 40 GPa and is little dependent on isotope substitution. We find that, consistently with the rather complex crystal structure of MH-III, the transition is much more convoluted in the hydrate than in ices VII and X. In methane hydrate, due to the presence of non equivalent O ions with distinct O-O distances, the dissimilar H bonds symmetrize in a pressure domain rather than at a single critical pressure.