This paper presents the two-dimensional sliding frictional contact between a rigid perfectly conducting flat punch and a functionally graded magneto-electro-elastic material (FGMEEM) layered half-plane. The electric potential and magnetic potential of the punch are assumed to be constant within the contact region. The magneto-electro-elastic (MEE) material properties of the FGMEEM layer vary as an exponential function along the thickness direction, and the Coulomb type friction is adopted within the contact region. By using the Fourier integral transform technique, the problem is reduced to coupled Cauchy singular integral equations of the first and second kinds for the unknown surface contact pressure, electric charge, and magnetic induction. An iterative method is developed to solve the coupled equations numerically and obtain the surface MEE fields. Then, the interior MEE fields are also obtained according to the surface MEE fields. Numerical results indicate that the gradient index and friction coefficient affect both the surface and interior MEE fields significantly.