In this study, we examine a generalized interval-valued multiobjective game with fuzzy constraint mappings (GIMGFCM). By employing interval support functions from convex geometry, the set of all $d$-dimensional interval vectors is embedded into the space of real-valued continuous functions defined on the unit sphere in ${\mathbb{R}}^d$. This embedding yields a closed convex cone $\mathcal{I}({\mathbb{R}}_{+}^d)$ within that function space. Using this cone, we define a partial order for interval vectors and establish the semi-continuity and generalized $\mathcal{I}({\mathbb{R}}_{+}^d)$-quasi-concavity of interval-vector-valued functions. On this basis, we propose a weak Pareto–Nash equilibrium concept for a GIMGFCM and prove an existence theorem for such equilibria. Finally, we apply the relevant theoretical framework to analyze a price competition problem between two firms that sell heterogeneous goods.