In this paper, a finite horizon H-infinity control problem is solved for a class of linear quantum systems using a dynamic game approach for the case of delayed measurements. The methodology adopted involves an equivalence between the quantum problem and an auxiliary classical stochastic problem. Then, the finite horizon H-infinity control problem for the class of linear quantum systems under consideration is solved for the case of delayed measurements by solving the finite horizon H-infinity control problem for an equivalent stochastic H-infinity control problem following a dynamic game approach.