Quasi-stationary Mean Field Games models consider agents who base their strategies on current information without forecasting future states. In this paper we address the first-order quasi-stationary Mean Field Games system, which involves an ergodic Hamilton-Jacobi equation and an evolutive continuity equation. Our approach relies on weak KAM theory. We introduce assumptions on the Hamiltonian and coupling cost to ensure continuity of the Peierls barrier and the Aubry set over time. These assumptions, though restrictive, cover interesting cases such as perturbed mechanical Hamiltonians.