We consider the Hamiltonian polynomial function H of degree fourth given by either H(x,y,{p_x},{p_y}) = \frac{1}{2}(p_x^2 + p_y^2) + \frac{1}{2}({x^2} + {y^2}) + {V_3}(x,y) + {V_4}(x,y),\,\,{\text{or}}\,H(x,y,{p_x},{p_y}) = \frac{1}{2}( - p_x^2 + p_y^2) + \frac{1}{2}( - {x^2} + {y^2}) + {V_3}(x,y) + {V_4}(x,y), where V3(x,y) and V4(x,y) are homogeneous polynomials of degree three and four, respectively. Our main objective is to prove the existence and stability of periodic solutions associated to H using the classical averaging method.