In this paper, we study the infinite-time mean field games with discounting, establishing an equilibrium where individual optimal strategies collectively regenerate the mean-field distribution. To solve this problem, we partition all agents into a representative player and the social equilibrium. When the optimal strategy of the representative player has the same feedback form as the strategy in the social equilibrium, we say that the system achieves a Nash equilibrium. We construct a Nash equilibrium using the stochastic maximum principle and infinite-time forward-backward stochastic differential equations (FBSDEs). By employing elliptic master equations, a class of distribution-dependent elliptic partial differential equations (PDEs), we provide a representation for the Nash equilibrium strategies. We prove the Yamada-Watanabe type theorem and show weak uniqueness for infinite-time FBSDEs. Furthermore, we prove that the solutions to a system of infinite-time FBSDEs can be employed to construct viscosity solutions for a class of distribution-dependent elliptic PDEs.