In this paper, the finite element method and the boundary element method are combined to solve numerically an exterior quasilinear elliptic problem. Based on an appropriate transformation and the Fourier series expansion, the exact qnasilinear artificial boundary conditions and a series of the corresponding approximations for the given problem are presented. Then the original problem is reduced into an equivalent problem defined in a bounded computational domain. We provide error estimate for the Galerkin method. Numerical results are presented to illustrate the theoretical results.