The emergence of graphene nanoplatelets (GPLs) has enabled the development of lightweight and high-strength plates, making it a prominent area of research in science and engineering. Therefore, it is essential to study the buckling performance of functionally graded graphene-reinforced composite (FG-GRC) plates. This paper presents a new meshless model to solve the buckling behavior problem of FG-GRC plates. The model is based on an improved Reddy-type third-order shear deformation theory (TSDT) with seven degrees of freedom and a moving Kriging (MK) interpolation method, which can overcome the challenge of implementing the second-type boundary conditions in meshless methods and eliminate the need for shear correction factors. The model is applicable to thin/medium/thick plate problems and has high computational accuracy. The Halpin-Tsai model is used to predict the effective Young's modulus of the FG-GRC plate, and the effective Poisson's ratio is determined using the mixture law. The meshless governing equation for the buckling of the FG-GRC plate with seven unknowns is derived based on the principle of minimum potential energy. The convergence and effectiveness of the proposed method are verified by comparing it with literature results. The numerical results demonstrate that when the total number of layers (N_L) of the FG-GRC plate is less than 10-15, the critical buckling load of the FG-O-type and FG-X-type plates changes more drastically than that of the epoxy pure plate, indicating that the stiffness of the graphene-reinforced plate decreases (or increases) rapidly in this stage, as opposed to the epoxy pure plate. However, when N_L exceeds 10-15, the change rate of the critical buckling load for the FG-GRC plate becomes smoother. Furthermore, the critical buckling load of the FG-GRC plate increases sharply when the length-thickness ratio of the GPLs reaches around 1000. Once the length-thickness ratio of GPLs surpasses 2000, the critical buckling load of the FG-GRC plate tends to stabilize, and the length-width ratio and length-thickness ratio of the GPLs have no significant effect on it. Overall, the research findings of this study not only contribute to the understanding of FG-GRC plates but also offer practical and insightful recommendations for their design and theoretical research.