Collisional absorption of dense, fully ionized plasmas in strong laser fields is investigated starting from a quantum kinetic equation with non-Markovian and field-dependent collision integrals in dynamically screened Born approximation. This allows to find rather general balance equations for the energy and the current. For high-frequency laser fields, quantum statistical expressions for the electrical current density and the cycle-averaged electron-ion collision frequency in terms of the Lindhard dielectric function are derived. The expressions are valid for arbitrary field strength assuming the nonrelativistic case. Numerical results are presented to discuss these quantities as a function of the applied laser field and for different plasma parameters. In particular, nonlinear phenomena such as higher harmonics generation and multiphoton emission and absorption in electron-ion collisions are considered. The significance to include quantum effects is demonstrated comparing our results for the collision frequency with previous results obtained from classical theories.