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The theoretical modeling of a functionally graded (FG) graphene nanoplatelet (GPL)-reinforced assembled beam–plate structure resting on elastic supports is presented for the first time, and its free vibration analysis is performed. Herein, the assembled structure is modeled according to the Kirchhoff plate theory and the Rayleigh beam theory. The graphene nanoplatelets (GPLs) gradiently distribute in the beam’s radial direction and in the plate’s thickness direction, respectively. By adopting the rule of mixture and the Halpin–Tsai model, the effective material properties can be obtained. By employing the Lagrange’s equation and considering the effects of Coriolis force and centrifugal force, the coupled governing equations of the assembled structure are determined. Furthermore, the assumed modes method and substructure modal synthesis method are applied to obtain the frequencies of the assembled beam–plate structure. A comprehensive numerical investigation is carried out to discuss the influence of the structural and material parameters on the vibration behavior of the beam–plate structure.