We consider a stabilized mixed finite element method introduced recently for the generalized Stokes problem. The method is obtained by adding suitable least squares terms to the dual-mixed variational formulation of the problem in terms of the velocity and the pseudostress. We obtain a new a posteriori error estimator of residual type and prove that it is reliable and locally efficient. Specifically, we develop an a posteriori error analysis based on the quasi-Helmholtz decomposition which allows us to prove the so-called local efficiency of the estimator with a non-homogeneous boundary condition. Finally, we present some numerical examples that confirm the theoretical properties of our approach.