Extensive attention has been devoted to the exploration of graphene owing to its remarkable mechanical characteristics and distinctive structural features. This study employs numerical methods to investigate the wave propagation behavior of defect-free single-layer graphene, incorporating its geometrically nonlinear properties by implementing second strain gradient elasticity. The primary focus is on capturing geometric nonlinearity through the introduction of nonlinear strain-displacement relationships. The paper expresses the Green-Lagrange strain, along with its first and second gradient, considering geometric nonlinearity. The governing equations and associated boundary conditions are confirmed using Hamilton's principle. Subsequently, the weak form is derived, incorporating element stiffness and mass matrices. To study two-dimensional wave propagation in graphene, the periodic structures' theory is applied, and the associated eigenvalue problems are solved. This approach provides new insights into various nonlinear dynamic properties of graphene.