In this paper, we propose a linear scalarization proximal point algorithm for solving lower semicontinuous quasiconvex multiobjective minimization problems. Under some natural assumptions and, using the condition that the proximal parameters are bounded, we prove the convergence of the sequence generated by the algorithm and, when the objective functions are continuous, we prove the convergence to a generalized critical point of the problem. Furthermore, for the continuously differentiable case we introduce an inexact algorithm, which converges to a Pareto critical point.