In this paper, we consider a class of infinite horizon forward and backward stochastic difference equations (FBSDEs) that are fully coupled. The adapted solution is given by establishing an explicit relation between the forward and backward components in terms of a generalized algebraic Riccati equation. The equivalence between the exponential stabilizability of the stochastic control system and the infinite horizon FBSDEs is derived. As an application, the FBSDEs are used to characterize the maximum principle of the infinite horizon stochastic optimal control problem.