In this work, we search how the production of gravitational waves is corrected by a spontaneous Lorentz violation triggered by the vacuum expectation value (VEV) of the bumblebee vector field ($b^\mu$). We consider the modified graviton equations of motion by the dynamics of the bumblebee field. In this formulation the graviton is still non-massive and has two degrees of freedom, but does not has a gauge transformation and the dispersion relation changes from $p^2 = 0$ to $p^2 + \xi (b^\mu p_\mu)^2 = 0$. Then, we solve these equations considering a current that acts as a source of the radiation. In this case, we apply the method of the Green's function to find a modified quadrupole formula for the perturbation.