Convolution and indirect meshless techniques are presented to solve the free vibration problem of irregular composite plates surrounded by a nonlinear elastic matrix with three parameters. Linear and parabolic foundations are examined. The displacement field is derived based on a first order transverse shear theory. In vibration study, the governing equation is reduced for the solution of the nonlinear eigenvalue problem and is solved by a direct iterative method. The influences of variable thickness, foundation stiffness, and supporting conditions on frequency values and mode shapes are investigated. The accuracy and efficiency of the two methods are carried out in two ways, firstly by comparison with exact, classical quadrature, and Rayleigh-Ritz approaches, and secondly by calculated the CPU time. Furthermore, some detailed results are presented to explore the effects of elastic and geometric properties of the vibrated irregular plate. From the results, it is found that the indirect moving least squares quadrature technique is an efficient scheme for vibration plates containing material discontinuity.