The stability of the transverse vibration of an axially moving functionally graded viscoelastic beam (FGVB) is studied based on the Euler beam theory. The complex frequencies of FGVB are firstly investigated by the differential quadrature method, where the main parameters such as axial speed and gradient index are considered. The effects of the axial speed and gradient index on the vibration characteristics and instable forms in supercritical region are also examined. Subsequently, the instability behaviors of FGVB with periodic perturbational velocity are addressed using the method of multiple scales and specific boundary conditions. The discussion is then mainly focused on the instable regions caused by summation resonance and subharmonic resonance, respectively, where the resonance occurs when the harmonic frequency approaches the sum or difference of any two mode natural frequencies. It is shown that the critical divergence and flutter velocities of the uniformly moving beam, as well as the stability regions of nonuniformly moving beam decrease with the increase of gradient index, and the viscoelastic effect becomes weaker with increasing gradient index. Moreover, the stability regions of axially moving FGVB for fully clamped boundary condition are larger than those for the simply supported case.