Functionally graded plates with cutouts are common load-bearing structures in engineering， making the study of their dynamic behavior crucial. For plates with regularly shaped cutouts， the symplectic superposition method combined with subdomain decomposition technique can be used to establish a comprehensive analytical solution framework. However， when the research object is extended to plates with arbitrarily shaped cutouts， which have broader applications， the analytical solution faces significant challenges due to the increased complexity of the boundary conditions. While approximate or numerical methods exist， they may suffer from time-consuming computations and strong mesh dependency， often leading to insufficient accuracy in results. A novel solution framework that integrates the symplectic superposition method with the transfer learning technique is established. It leverages the analytically solvable natural frequencies of functionally graded rectangular plates with rectangular cutouts for knowledge transfer， thereby achieving efficient and highly accurate solutions for the free vibration problems of functionally graded plates with arbitrary quadrilateral cutouts. First， a multi-layer perceptron neural network is pre-trained on large-scale analytical solution dataset to extract the complex mapping relationships of geometric parameters between different shapes. Second， the pre-trained model is transferred based on few-shot finite element simulation data， effectively transferring knowledge from the natural frequency dataset of the rectangular plates with rectangular cutouts to the free vibration of plates with arbitrary quadrilateral cutouts. Finally， the accuracy and applicability of the proposed solution framework for predicting the natural frequencies of functionally graded plates with arbitrary quadrilateral cutouts are validated， using metrics such as mean squared error and coefficient of determination. The transfer learning technique is used to leverage small-sample data and existing analytical solutions to efficiently and accurately solve free vibration problems of functionally graded with arbitrary quadrilateral plates cutouts. The proposed framework， adaptable to various working conditions through model parameter adjustments， offers a novel strategy for the mechanical analysis of complex-shaped plates.