We report the temperature and pressure (p_max ≃ 1.5 GPa) evolution of the crystal structure of the Weyl semimetal T_d-MoTe₂ by combination of neutron diffraction and the X-ray total scattering experiments. We find that the fundamental non-centrosymmetric structure T_d is fully suppressed and transforms into a centrosymmertic 1T′ structure at a critical pressure of p_cr ∼ 1.2-1.4 GPa. This is strong evidence for a pressure induced quantum phase transition (QPT) between topological to a trivial electronic state. Although the topological QPT has strong effect on magnetoresistance, it is interesting that the superconducting (SC) critical temperature T_c, the superfluid density, and the SC gap all change smoothly and continuously across p_cr and no sudden effects are seen concomitantly with the suppression of the T_d structure. This implies that the T_c, and thus the SC pairing strength, is unaffected by the topological QPT. However, the QPT requires the change in the SC gap symmetry from non-trivial s⁺⁻ to a trivial s⁺⁺ state, which we discuss in this work. Our systematic characterizations of the structure and SC properties associated with the topological QPT provide deep insight into the pressure induced phase diagram in this topological quantum material.