The generalization of a new numerical approach with simultaneous space-time finite element discretization for viscoelastic problems developed in the papers by Buch et al. and Idesman et al. is presented for the case of the generalized viscoelastic Maxwell model. The approach has some very attractive advantages with respect to semidiscretization methods. Numerical examples show the effectiveness of simultaneous space-time finite element calculations and a high convergence rate for adaptive refinement.