In this paper, we investigate the generalized Stieltjes-Wigert and q-Laguerre polynomials. We derive the second- and the third-order nonlinear difference equations for the subleading coefficients of these polynomials and use them to find a few terms of the formal expansions in powers of q. We also show how the recurrence coefficients in the three-term recurrence relation for these polynomials can be computed efficiently by using the nonlinear difference equations for the subleading coefficient. Moreover, we obtain systems of difference equations with one of the equations being q-discrete Painlevé III or V equations and analyze them by a singularity confinement. We also discuss certain generalized weights.