You talk about morphisms from $\Delta_\bullet$ to a graph $G$. I presume a morphism from $\Delta_n$ to $G$ is just an embedding of $\Delta_n$ in $G$, that is an $(n+1)$-clique in $G$ with a labelling of its vertices from $0$ to $n$. It seems to me that defining (co)homology in this way will be the same as the standard simplicial/singular (co)homology of the space $\widehat G$ obtained from $G$ by "filling in" each $(n+1)$-clique with an $(n+1)$-simplex.
As a example consider the graph $G$ consisting of the vertices and edges of a regular icosahedron. Then its $3$-cliques correspond to the faces of the icosahedron and it has no $4$-cliques. Therefore $\widehat G$ will consist of the boundary of the solid icosahedron in $\mathbb{R}^3$ and so your second homology group will be nonzero.
I hasten to add that I have not checked any details, and admit in advance that my thoughts here may be msiguided or just complete nonsense.