Abstract: The transverse nonlinear vibration of the axial moving system has become one of the hot subjects at home and abroad. At present, most of the related studies are on the homogeneous boundary conditions. However, the nonhomogeneous boundary conditions are more common in the engineering practice. There are relatively few publications on the axial moving system with homogeneous boundary conditions. In order to study the effect of nonhomogeneous boundary conditions on the transverse nonlinear vibration of an axial moving beam, the nonlinear parametric vibrations of an axially accelerating viscoelastic Euler beam under nonhomogeneous boundary conditions are studied in this paper. The variable tension caused by the axial acceleration is introduced. A nonlinear integro-partial-differential equation and corresponding nonhomogeneous boundary conditions of an axially accelerating viscoelastic beam are presented. The effects of nonhomogeneous boundary conditions are highlighted. The method of multiple scales is used to establish the solvability conditions. The steady-state response of the beam was obtained from the solvability condition. According to the Routh-Hurvitz criterion, the stability of the response was determined. Some numerical examples are introduced to demonstrate the effect of the viscoelastic coefficient, mean speed, axial speed fluctuation amplitude, nonlinear coefficient, and the nonhomogeneous boundary conditions on the steady-state response. The larger viscoelastic coefficient leads to the smaller instability interval of the trivial solutions and the smaller stable steady-state response amplitude especially for the second mode. The instability interval and the stable steady-state response amplitude with the nonhomogeneous boundary conditions are larger than that with the homogeneous boundary conditions. The differential quadrature scheme is introduced to confirm the approximate analytical results. The numerical results show reasonable agreement with the approximate analytical results.