The static and dynamic critical properties of the ferromagnetic q-state Potts models on a square lattice with q = 2 and 3 are numerically studied via the nonequilibrium relaxation method. The relaxation behavior of both the order parameter and energy as well as that of the second moments are investigated, from which static and dynamic critical exponents can be obtained. We find that the static exponents thus obtained from the relaxation of the order parameter and energy together with the second moments of the order parameter exhibit a close agreement with the exact exponents, especially for the case of q = 2 (Ising) model, when care is taken in the choice of the initial states for the relaxation of the second moments. As for the case of q = 3, the estimates for the static exponents become less accurate but still exhibit reasonable agreement with the exactly known static exponents. The dynamic critical exponent for q = 2 (Ising) model is estimated from the relaxation of the second moments of the order parameter with mixed initial conditions to give z(q = 2) = 2.1668(19). ; Comment: 29 pages, 16 figures