In this study, we develop a high-order accurate discontinuous Galerkin scheme using curvilinear quadrilateral elements for solving elliptic interface problems. To maintain accuracy for curvilinear quadrilateral elements with $Q^k$-polynomial basis functions, we select Legendre-Gauss-Lobatto quadrature rules with $k+2$ integration points, including end points, on each edge. Numerical experiments show quadrature rules are appropriate and the numerical solution converges with the order $k+1$. Moreover, we implement the Uzawa method to rewrite the original linear system into smaller linear systems. The resulting method is three times faster than the original system. We also find that high-order methods are more efficient than low-order methods. Based on this result, we conclude that under the condition of limited computational resources, the best approach for achieving optimal accuracy is to solve a problem using the coarsest mesh and local spaces with the highest degree of polynomials.