In this paper, we propose a high order multiquadric trigonometric quasi-interpolation method for function approximation and derivative approximation based on periodic sampling data. Moreover, we apply it to solve time-dependent nonlinear partial differential equations (PDEs) with periodic solutions. Learning from the construction of the trigonometric B-spline quasi-interpolation, we derive the new quasi-interpolation scheme by replacing the truncated trigonometric polynomial with a high degree, infinitely smooth multiquadric trigonometric kernel. By appropriately choosing the shape parameter in the kernel, we prove that the proposed method achieves higher order of convergence than the existing multiquadric trigonometric quasi-interpolation method. Finally, we conduct extensive numerical experiments to demonstrate the accuracy and efficiency of the proposed method for approximating unknown function and solving different types of PDEs including the one-dimensional KdV equation and two-dimensional Allen-Cahn equation. [ABSTRACT FROM AUTHOR]