The two-dimensional BEG model with nearest neighbor bilinear and positive biquadratic interaction is simulated on a cellular automaton, which is based on the Creutz cellular automaton for square lattice. Phase diagrams characterizing phase transitions of the model are presented for comparison with those obtained from other calculations. We confirm the existence of the tricritical points over the phase boundary for D / K > 0. The values of static critical exponents (alpha, beta, gamma and nu) are estimated within the framework of the finite size scaling theory along D / K = -1 and 1 lines. The results are compatible with the universal Ising critical behavior except the points over phase boundary.