We present an investigation of the interquark potential determined from the qq̄ Bethe-Salpeter (BS) amplitude for heavy quarkonia in lattice QCD. The qq̄ potential at finite quark mass mq can be calculated from the equal-time and Coulomb gauge BS amplitude through the effective Schrödinger equation. The definition of the potential itself requires information about a kinetic mass of the quark. We then propose a self-consistent determination of the quark kinetic mass on the same footing. To verify the proposed method, we perform quenched lattice QCD simulations with a relativistic heavy-quark action at a lattice cutoff of 1/a≈2.1GeV in a range 1.0≤mq≤3.6GeV. Our numerical results show that the qq̄ potential in the mq→∞ limit is fairly consistent with the conventional one obtained from Wilson loops. The quark-mass dependence of the qq̄ potential and the spin-spin potential are also examined.