Nanoplate structures are widely used in nanoelectromechanical systems because of their excellent mechanical properties. At the same volume, the specific surface area of the nano-laminates is much larger than that of the single-layer nanomaterials, and the surface effect is more significant. The influence of the surface effect on nanostructures can be regarded as the combination of surface elasticity and surface residual stress. The classical plate-shell theory does not consider the surface effect and is no longer suitable for describing nanostructures such as nanoplate-shells. As a new type of composite materials, functionally graded materials have received more and more attention from researchers. The mechanical properties of micro/nano structural components made of functionally graded materials are completely different from the macroscopic structures made of conventional materials. Plate structure is a basic component in nanoelectromechanical systems, so it is necessary to study the mechanical properties of plates made of functionally graded materials. In this paper, based on Kirchhoff plate theory and Mindlin plate theory considering shear deformation, the buckling and post-buckling behaviors of functionally graded sandwich nanoplates with surface effect are studied. Based on the force balance analysis, the governing equations of buckling and post-buckling are obtained. The analytical solutions of critical buckling loads under uniaxial and biaxial compression are given. By using the Galerkin method, the approximate solutions of critical post-buckling loads under movable and immovable boundary conditions are given. Numerical results show that the influence of surface effects on the stability of functionally graded nano-laminated plates is related to the volume fraction of the materials that make up the plates, as well as to the ratio of structural surface area to volume. Considering that shear deformation will reduce the critical load for buckling and post-buckling of functionally graded nano-laminated plates, the influence of shear deformation can be ignored for thinner nano-laminated plates.