Out-of-phase (OOP) and in-phase (IP) quadrupole beams propagation in self-focusing photoinduction tetragonal photonic lattices is investigated by alternative direction implicit-beam propagation method (ADI-BPM). The evolution of OOP and IP quadrupole beams in continuous homogeneous medium, self-focusing medium and self-focusing photoinduction photonic lattices is presented respectively. In the continuous homogeneous medium, it is shown that OOP quadrupole beams are linearly diffracted, with the peak intensity decreasing gradually. Besides, the beams repel each other and the peak spacing becomes larger. IP quadrupole beams are diffracted originally, and then fused into the single beam. The peak is decreased in the beginning and then increased. As the beam is transmitted forward and diffracted, the peak intensity declines. In the self-focusing nonlinear medium, the OOP quadrupole beams could self-focus and the separation becomes more obvious, while the IP quadrupole beams could attract each other and merge into a single beam. In the self-focusing light photoinduction lattices, under appropriate conditions, the OOP quadrupole beams could form the stable OOP quadrupole solitons whose diagonal humps have the same phase but the adjacent humps have π-phase difference, while the IP quadrupole beams could form the localized IP quadrupole solitons whose four humps have the same phase.