We report magnetic and thermodynamic properties of a 4d1 (Mo5+) magnetic insulator MoOPO4 single crystal, which realizes a J1−J2 Heisenberg spin-1/2 model on a stacked square lattice. The specific-heat measurements show a magnetic transition at 16 K which is also confirmed by magnetic susceptibility, ESR, and neutron diffraction measurements. Magnetic entropy deduced from the specific heat corresponds to a two-level degree of freedom per Mo5+ ion, and the effective moment from the susceptibility corresponds to the spin-only value. Using ab initio quantum chemistry calculations, we demonstrate that the Mo5+ ion hosts a purely spin-1/2 magnetic moment, indicating negligible effects of spin-orbit interaction. The quenched orbital moments originate from the large displacement of Mo ions inside the MoO6 octahedra along the apical direction. The ground state is shown by neutron diffraction to support a collinear Néel-type magnetic order, and a spin-flop transition is observed around an applied magnetic field of 3.5 T. The magnetic phase diagram is reproduced by a mean-field calculation assuming a small easy-axis anisotropy in the exchange interactions. Our results suggest 4d molybdates as an alternative playground to search for model quantum magnets.