In this talk, we introduce a new infinite horizon domination-monotonicity framework. In this framework, by the method of continuation and some subtle techniques, we obtain an existence and uniqueness result and a pair of estimates for the solutions to a kind of infinite horizon coupled forward-backward stochastic differential equations (FBSDEs, for short). Then, the theoretical result of FBSDEs is applied to solve a stochastic linear-quadratic (LQ, for short) optimal control problem with random time-varying coefficients on infinite horizon. The unique open-loop optimal control is characterized by the solution of an infinite horizon FBSDE. Moreover, we find and illustrate a different phenomenon between the LQ problems on infinite horizon and finite horizon.