In this paper we introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for numerically solving a class of nonlinear Stokes models arising in quasi-Newtonian fluids. Similarly as in previous papers dealing with the application of mixed finite element methods to these nonlinear models, we use the incompressibility condition to eliminate the pressure, and set the velocity gradient as an auxiliary unknown. In addition, we enrich the HDG formulation with two suitable augmented equations, which allows us to apply known results from nonlinear functional analysis, namely a nonlinear version of Babuka-Brezzi theory and the classical Banach fixed-point theorem, to prove that the discrete scheme is well-posed and derive the corresponding a priori error estimates. Then we discuss some general aspects concerning the computational implementation of the method, which show a significant reduction of the size of the linear systems involved in the Newton iterations. Finally, we provide several numerical results illustrating the good performance of the proposed scheme and confirming the optimal order of convergence provided by the HDG approximation.