For a quiver $Q$, we take $\mathcal{M}$ an associated toric Nakajima quiver variety and $Γ$ the underlying graph. In this article, we give a direct relation between a specialisation of the Tutte polynomial of $Γ$, the Kac polynomial of $Q$ and the Poincaré polynomial of $\mathcal{M}$. We do this by giving a cell decomposition of $\mathcal{M}$ indexed by spanning trees of $Γ$ and `geometrising' the deletion and contraction operators on graphs. These relations have been previously established by Sturmfels-Hausel and (Crawley-Boovey)-Van den Bergh, however the methods here are more hands-on.