An iron-based superconductor LaFeAsO_1−xH_x(0≤x≤0.6) undergoes two antiferromagnetic (AF) phases upon H doping. We investigated the second AF phase (x=0.6) using NMR techniques under pressure. At pressures up to 2 GPa, the ground state is a spin-density-wave state with a large gap; however, the gap closes at 4.0 GPa, suggesting a pressure-induced quantum critical point. Interestingly, the gapped excitation coexists with gapless magnetic fluctuations at pressures between 2 and 4 GPa. This coexistence is attributable to the lift up of the d_xy orbital to the Fermi level, a Lifshitz transition under pressure.