Determining feedback Nash equilibrium solutions of nonzero-sum dynamic games is generally challenging. In this paper, we propose four different iterative algorithms to find Nash equilibrium strategies for discrete-time linear quadratic games. The strategy update laws are based on the solution of either Lyapunov or Riccati equations for each player. Local convergence criteria are discussed. Motivated by the fact that in many practical scenarios each player in the game may have access to different (incomplete) information, we also introduce purely data-driven implementations of the algorithms. This allows the players to reach a Nash equilibrium solution of the game via scheduled experiments and without knowledge of each other's performance criteria or of the system dynamics. The efficacy of the presented algorithms is illustrated via numerical examples and a practical example involving human-robot interaction.