This study presents the effectiveness of novel fractional modeling based on differential quadrature shape functions (i.e., Lagrange interpolation polynomial, Cardinal sine function, Delta Lagrange kernel, and Regularized Shannon kernel) in solving the nonlinear vibration of multilayered piezoelectric plates with nonlinear elastic support. A piezoelectric composite laminated plate is investigated while resting on nonlinear Winkler and Visco-Pasternak elastic foundations. Based on 3D elasticity theory and principles of piezoelectricity, the governing equations of motion are derived. Differential quadrature methods based on four shape functions, in conjunction with the Generalized Caputo approach, are presented as numerical techniques for solving this problem. The perturbation method is implemented for solving the obtained nonlinear eigenvalue problem. A MATLAB code is written for each technique to solve this problem and extract the numerical results. To validate these techniques, the computed results are compared with the previous published exact results. Additionally, a parametric analysis is offered to discuss the influence of length to thickness ratio, foundation parameters, damping parameter, and different boundary conditions on the frequencies and mode shapes. Therefore, it is concluded that the computed results through the proposed schemes may be effectively utilized in structural health monitoring.