I can at least propose a definition. To my mind, the categorically best behaved category of graphs is the category of presheaves on $\{ \bullet \rightrightarrows \bullet \}$ ("directed multigraphs"); explicitly, it's the category of tuples $(E, V, s, t)$ of an edge set $E$, a vertex set $V$, and two functions $s, t : E \to V$ (the source and target of an edge). Among other things, this is the category over which the category of small categories is monadic. So:
Proposal: A "group graph" is a group object in the category of graphs (in the above sense).
Explicitly, a group graph is a tuple $(E, V, s, t)$ of two groups $E, V$ (so edges admit a multiplication too, not just vertices) and two group homomorphisms $s, t : E \to V$. Note that this is the same thing as a graph object in the category of groups, and that another way to say "graph object" is "coequalizer diagram."
Said another way, first we need a group operation $\otimes$ on vertices, and then given two edges $e_1 : s_1 \to t_1, e_2 : s_2 \to t_2$, they admit a product which takes the form $e_1 \otimes e_2 : s_1 \otimes s_2 \to t_1 \otimes t_2$, and this product also satisfies the group axioms.
Example. Given any homomorphism $f : G \to H$ of groups, taking its kernel pair gives a diagram
$$G \times_H G \rightrightarrows G \to H$$
and the first part $G \times_H G \rightrightarrows G$ of this diagram (where the two morphisms are the two projections) is naturally a graph object. This construction generalizes to any category with pullbacks.
A related but more familiar thing you could ask for is a "group object" in categories rather than graphs; that is, a monoidal category in which every object is invertible (this is the motivation for the notation $\otimes$). This notion and various related notions are important in algebraic topology and other fields which make use of higher category theory; compare the notion of 2-group and "Picard groupoid."
Another related but more familiar thing you could ask for is a simplicial group (thinking of graphs as being like simplicial sets, but over a truncated version of the simplex category).
