This study details a nonlinear buckling and post-buckling analysis of Functionally Graded Material (FGM) plates, which are increasingly utilized in advanced engineering applications like aerospace due to their superior thermal and mechanical properties. The design and integrity of these structures under complex loading conditions, however, pose significant challenges, particularly regarding their stability under compressive forces.To address this, the structural formulation is based on the First-Order Shear Deformation Theory (FSDT), which is well-suited for capturing shear deformation effects that become significant in thin to moderately thick structures. By minimizing the total potential energy of the system, a comprehensive mathematical model is derived. The subsequent solution of the highly complex nonlinear governing equations is achieved through an innovative hybrid numerical approach. This method leverages the robust Asymptotic Numerical Method (ANM), a powerful continuation technique known for its ability to efficiently trace complex equilibrium paths, and integrates it with the versatile Finite Element Method (FEM). This combination effectively handles the spatial discretization of the plate while providing a stable and accurate means to track the structural behavior beyond the critical buckling point.The presented approach is particularly effective in accurately identifying critical buckling loads and precisely locating bifurcation points along the equilibrium paths. The method's effectiveness is demonstrated by its high computational efficiency and accuracy, which are critical for the design and safety analysis of FGM components. The findings of this research provide a reliable tool for structural engineers to analyze the stability and behavior of FGM plates, contributing to the development of more resilient and efficient structural designs.