This paper used the differential quadrature method (DQM) to investigate the dynamic behaviors for the nonplanar nonlinear vibrations of an axially moving viscoelastic beam. The mathematical model of the non-planar nonlinear vibration of the axially moving viscoelastic beam is a very complex nonlinear partial differential equations. Firstly, the nonlinear governing equations of nonplanar motion for the axially moving viscoelastic beam are solved by using the DQM for the first time. Then, based on the numerical results, the bifurcation diagrams, phase portraits and time history diagrams are used to analyze the complex nonlinear dynamic behaviors of the axially moving viscoelastic beam.