Combining the spectral geometry method and the incremental harmonic balance method (IHB), the nonlinear analytical model of functionally graded graphene platelet reinforced composite (FG-GPLRC) beams is developed in this paper. Nonlinear free and forced vibration characteristics are studied based on this model containing viscoelastic foundation and geometric nonlinear factors. Geometric nonlinear strain-displacement relationship and Lagrangian energy generalization for the FG-GPLRC beam structure are derived according to the Von-Kármán and Timoshenko theories. The IHB method is applied to track the nonlinear vibration response solution of the reduced-order model, which is constructed by introducing linear modal components into the overall nonlinear dynamic equation of the FG-GPLRC beam. Through comparison of the present solution with those from numerical method and literature, the correctness of the established nonlinear analytical model is evidenced. Further, the influence of material-, geometry-, foundation-related parameters and boundary constraints on the nonlinear frequency parameter and amplitude-frequency response of FG-GPLRC beams rested on viscoelastic foundation is analyzed, which can serve as a theoretical guide for designing and evaluating the dynamic environment adaptation of FG-GPLRC beams.