We present the moment cone relaxation for a class of polynomial optimization problems (POPs) to extend the results on the completely positive cone programming relaxation for the quadratic optimization (QOP) model by Arima, Kim and Kojima. The moment cone relaxation is constructed to take advantage of sparsity of the POPs. We establish the equivalence between the optimal value of the POP and that of the moment cone relaxation under conditions similar to the ones assumed in the QOP model. For tractable numerical methods, the doubly nonnegative cone relaxation is derived from the moment cone relaxation. Exploiting sparsity in the doubly nonnegative cone relaxation and its incorporation into Lasserre's semidefinite relaxation are briefly discussed.