Dark energy is the dominant component of the energy density in the Universe. In a previous paper, we have shown that the collapse of dark energy fields leads to the formation of supermassive black holes with masses comparable to the masses of black holes at the centers of galaxies. Thus, it becomes a pressing issue to investigate the other physical consequences of the collapse of dark energy fields. Given that the primary interactions of dark energy fields with the rest of the Universe are gravitational, it is particularly interesting to investigate the gravitational wave signals emitted during the collapse of dark energy fields. This is the focus of the current work described in this paper. We describe and use the 3+1 BSSN formalism to follow the evolution of the dark energy fields coupled with gravity and to extract the gravitational wave signals. Finally, we describe the results of our numerical computations and the gravitational wave signals produced by the collapse of dark energy fields.