Materials belonging to the graphene family are two-dimensional staggered monolayers that undergo topological phase transitions under the influence of an external electric field or off-resonant optical field. Inspired by the interplay between topological matter and the helicity of photons, we investigate various topological quantum phases of the graphene family materials (GFMs), when subject to an external electric field and irradiated by off-resonant light. Using the Kubo formalism, we derive analytic expressions of the valley and spin-resolved conductivities of silicene. We then show that the topological quantum phase transitions can be modulated by an external electric field or irradiating circularly polarized light on the surface. Based on a general beam propagation model, we theoretically investigate the transitional Kerr rotations in silicene in different phases. Our results identify topological phases where Kerr rotations and ellipticity can be maximized. We believe that our results are helpful for developing novel practical devices based on the Kerr effect of silicene.