In the past few decades, strategies for designing new two-dimensional covalent organic framework (2D-COF) structures have been limited to the shape of positive hexagonal pores, and the underlying relationship between their structure and electronic properties still remains unclear. Herein, novel 2D-COFs with C, N and H elements confined to the quadrilateral-pore skeleton based on first-principles calculations and the topological assembly of different benzene-based building blocks were designed and studied. These 2D-COFs enriched the topology types and can offer an ideal platform for band engineering aimed at spontaneously driving the hydrogen evolution reaction (HER) under visible light irradiation. The approach for regulating pore structures on nodes, linkers and linkages can effectively tune band gaps, and thus the 2D-COF, consisting of benzene building blocks and imine linkages, has the optimal activity for the photocatalytic HER under common visible light conditions. Furthermore, the integrated pz-orbital population was found to evaluate the photocatalytic activity efficiently. We demonstrate that the pz-orbital population is in linear relationship with the intensity of H+ adsorption, indicating that the total contribution of the pz-orbital electrons can be an efficient descriptor for screening suitable 2D-COF structures for use as photocatalysts for the HER. Therefore, this work presents a new strategy for designing novel quadrilateral-pore 2D-COFs as visible-light photocatalysts and provides an important insight into the relationship between catalytic activity and the population of activated electrons.