<p>This paper is concerned with fully coupled forward–backward stochastic differential equations (FBSDEs, for short) with non-zero terminals in infinite horizon. By introducing stochastic Lipschitz conditions and constructing infinite horizon domination–monotonicity conditions, the well-posedness of this kind of infinite horizon FBSDEs including the existence, uniqueness and a pair of estimates is proved. Moreover, the theoretical results are applied to solve four kinds of linear-quadratic (LQ, for short) stochastic optimal control problems with random coefficients in infinite horizon. Due to the unboundedness and randomness of coefficients, the results of the FBSDEs and LQ problems obtained in this paper, even if they are degenerated to finite horizon, contain more situations than the results in the literature.</p>