We present a microscopic calculation of the nondissipative current through a superconducting quantum point contact coupled to a mechanical oscillator. Using the non-equilibrium Keldysh Green function approach, we determine the current-phase relation. The latter shows that at certain phases, the current is sharply suppressed. These dips in the current-phase relation provide information about the oscillating frequency and coupling strength of the mechanical oscillator. We also present an effective two-level model from which we obtain analytical expressions describing the position and width of the dips. Our findings are of relevance for nanomechanical resonators based on superconducting materials.