In this paper, we introduce the generalized Ekeland’s variational principle in several forms. The general setting of our results includes a graphical metric structure and also employs a generalized -distance. We then applied the proposed variational principles to obtain existence theorems for a class of quasi-equilibrium problems whose constraint maps are induced from the graphical structure. The conditions used in our existence results are based on a very general concept called a convergence class. Finally, we deduce the existence of a generalized Nash equilibrium via its quasi-equilibrium reformulation. A validating example is also presented.