In this paper, we consider a class of non-cooperative N-player finite-horizon linear-quadratic dynamic games with linear constraints. We introduce a constrained feedback information structure and provide necessary and sufficient conditions for the existence of a constrained feedback Nash equilibrium. For this class of games, we show that the constrained feedback Nash equilibrium can be obtained from a parametric feedback Nash equilibrium associated with a unconstrained parametric linear-quadratic game where the parameters are chosen in a specific way. Further, we show that this relation leads to a fixed-point interpretation. Further, under a few assumptions, we show that these fixed points can be obtained as solutions of a single large-scale linear-complementarity problem, thereby providing a method to compute the constrained feedback Nash equilibria. We illustrate our results with a numerical example.