An efficient numerical scheme is developed to solve the system of multi-dimensional nonlinear Fredholm integral equations (MDNFIEs) of the second kind in terms of the variable transformation technique of Sidi type in conjunction with the trapezoidal quadrature rule. Using the product Nyström method, the considered integral system is discretized into a set of nonlinear algebraic equations. Additionally, a rigorous convergence analysis of the proposed method is provided, thus demonstrating that compared with the classical trapezoidal quadrature rule approach, this scheme achieves a significantly improved convergence rate. Furthermore, numerical examples are presented to validate the efficiency and accuracy of the described method.