In this paper, a subgradient projection method for quasiconvex minimization problems is provided. By using strong subdifferentials, it is proved that the generated sequence of the proposed algorithm converges to the solution of the minimization problem of a proper, lower semicontinuous and strongly quasiconvex function (in the sense of Polyak [18]) under the same assumptions than for convex functions with the convex subdifferential. Furthermore, a quasi-linear convergence rate of the iterates, which extends similar results for the general quasiconvex case, is also provided.