Introduced in the 1970's by Martinet for minimizing convex functions and extended shortly afterwards by Rockafellar towards monotone inclusion problems, the proximal point algorithm turned out to be a viable computational method for solving various classes of optimization problems, in particular with nonconvex objective functions. We propose first a relaxed-inertial proximal point type algorithm for solving optimization problems consisting in minimizing strongly quasiconvex functions whose variables lie in finitely dimensional linear subspaces. The method is then extended to equilibrium problems where the involved bifunction is strongly quasiconvex in the second variable. Possible modifications of the hypotheses that would allow the algorithms to solve similar problems involving quasiconvex functions are discussed, too. Numerical experiments confirming the theoretical results, in particular that the relaxed-inertial algorithms outperform their ``pure'' proximal point counterparts, are provided, too.