We study the Stokes-Poisson-Boltzmann equations with mixed Dirichlet and Navier boundary conditions. The system consists of the incompressible Stokes equations coupled with a nonlinear Poisson-Boltzmann equation through electrostatic forcing and convective transport effects. To handle the Navier boundary conditions in a unified framework, we employ Nitsche`s method for their weak imposition within a conforming finite element setting. We derive a consistent and stable discrete formulation and establish the well-posedness of the resulting problem. By carefully choosing the penalty parameters, the bilinear form is shown to be coercive and continuous. A priori error estimates are proved in the natural energy norms, yielding optimal-order convergence under suitable regularity assumptions. Furthermore, we develop residual-based a posteriori error estimators that incorporate element residuals, inter-element jump terms, and boundary contributions arising from the Nitsche formulation. The estimators are shown to be reliable and locally efficient, providing a rigorous basis for adaptive mesh refinement. Numerical experiments confirm the theoretical results and demonstrate the robustness and accuracy of the proposed method for the Stokes-Poisson-Boltzmann system.