We discuss a subgradient projection method for dealing with the nonconvex nonsmooth multiobjective optimization problem when every component of the vector-valued function is strongly quasiconvex in the sense of Polyak [Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Soviet Math. 1966;7:72-75]. Under mild assumptions, we ensure that the generated sequence converges to an efficient solution point of the multiobjective optimization problem and we provide a kind of linear convergence rate. The algorithm is based on a specific generalized subdifferential that was recently introduced for strongly quasiconvex functions, this approach provides new and valuable information on the generated sequence and its convergent point. Finally, we present numerical experiments for classes of quadratic fractional programming problems.