In this research, a novel size-dependent meshfree method is proposed within the framework of the nonlocal strain-gradient theory in order to evaluate the buckling behavior of functionally graded metal–semiconductor nanoplates reinforced with graphene nanoplatelets, while the nanoplates with elastically restrained edges is placed on a Winkler–Pasternak elastic foundation. The presented model simultaneously accounts for the effects of nonlocal stiffness and strain gradient, thereby covering the intrinsic softening and stiffening mechanisms at the nanoscale. The governing equations are derived using the principle of minimum total potential energy and are discretized through the moving Kriging meshfree method. This method effectively resolves the higher-order derivatives appearing in the nonlocal strain-gradient theory. The mechanical properties corresponding to each layer of the plate along the thickness, which is reinforced by graphene nanoplatelets, are determined through the modified Halpin–Tsai micromechanical model together with the rule of mixtures. Comparison of the results obtained from the proposed method with available analytical and numerical approaches confirms the accuracy and computational efficiency of the proposed method. Furthermore, a parameter-based investigation is carried out to clarify the effects of material gradation, graphene weight fraction, nonlocal and strain-gradient parameters, circular cutout size, the stiffness of the elastic foundation and stiffnesses of the elastically restrained edges on the buckling responses of functionally graded graphene-reinforced nanoplates.