In this article we consider the a priori and a posteriori error analysis of two-grid $hp$-version discontinuous Galerkin finite element methods for the numerical solution of a strongly monotone quasi-Newtonian fluid flow problem. The basis of the two-grid method is to first solve the underlying nonlinear problem on a coarse finite element space; a fine grid solution is then computed based on undertaking a suitable linearization of the discrete problem. Here, we study two alternative linearization techniques: the first approach involves evaluating the nonlinear viscosity coefficient using the coarse grid solution, while the second method utilizes an incomplete Newton iteration technique. Energy norm error bounds are deduced for both approaches. Moreover, we design an $hp$-adaptive refinement strategy in order to automatically design the underlying coarse and fine finite element spaces. Numerical experiments are presented which demonstrate the practical performance of both two-grid discontinuous Galerkin methods.