Graphene is a two-dimensional material with single-layer honeycomb lattice structure formed by sp² hybrid connection of carbon atoms. Graphene has excellent optical, electrical, thermal and mechanical properties, and it is considered to be an ideal material for future flexible optoelectronic devices. In recent years, the nonlinear optical properties and regulation of graphene nanostructures have attracted experimental and theoretical interest. Graphene has good delocalization of π-electrons and its unique plane structure, showing good nonlinear optical properties. Graphene quantum dots can be regarded as small graphene nanoflakes. Their unique electronic structure is closely related to the non-bond orbitals on the boundary/edge. Therefore, it is very important to study the boundary/edge effect on the electronic and optical properties of nanographene. In this paper, effects of the number of edge C=C double bonds and Borazine (B₃N₃) doping on the nonlinear optical properties and UV-Vis absorption spectrum of graphene quantum dots are studied by the quantum chemical calculation methods, respectively. It is found that the symmetry of hexagonal graphene quantum dots decreases and the symmetry of charge distribution is broken when C=C double bond is introduced into the armchair edge, which leads the second-order nonlinear optical activity to be enhanced. During the transition from armchair to zigzag edge, the polarizability and the second hyperpolarizability of hexagonal graphene quantum dots and B₃N₃-doped graphene quantum dots increase linearly with the number of introduced C=C double bonds incrrasing. In addition, the edge also has an important influence on the absorption spectrum of graphene quantum dots. For graphene quantum dots and B₃N₃-doped graphene quantum dots, the introduction of C=C double bond at the armchair edge increases the highest occupied molecular orbital energy level and also reduces the lowest unoccupied molecular orbital energy level, which reduces the energy gap between the frontier molecular orbitals, and thus resulting in the red-shift of the maximum absorption wavelength. The doping of B₃N₃ ring will increase the energy gap between molecular frontier orbitals of graphene quantum dots, leading the UV-Vis absorption spectrum of graphene quantum dots to be blue-shifted. This study provides theoretical guidance for controlling the nonlinear optical response of graphene quantum dots by edge modification.