By using the gravity equation for quantum mechanical systems that takes into account the non-local interaction of the quantum potential the paper derives the quantization of the graviton in the limit of weak gravity. The output of the theory shows that, in the non-Minkowskian quantum approach proposed, where the coupling between the gravitational equation and the field is explicitly defined, a massless boson field can be associated to the graviton. The paper shows that the commutation relations of the metric tensor of the gravitational waves can be analytically derived by the quantization of the associated graviton field.