The Pauli groups are ubiquitous in quantum information theory because of their usefulness in describing quantum states and operations and their readily understood symmetry properties. In addition, the most well-understood quantum error correcting codes – stabilizer codes – are built using Pauli operators. The eigenstates of these operators – stabilizer states – display a structure (e.g., mutual orthogonality relationships) that has made them useful in examples of multi-qubit non-locality and contextuality. Here, we apply the graph-theoretical contextuality formalism of Cabello, Severini and Winter to sets of stabilizer states, with particular attention to the effect of generalizing two-level qubit systems to odd prime d-level qudit systems. While state-independent contextuality using two-qubit states does not generalize to qudits, we show explicitly how state-dependent contextuality associated with a Bell inequality does generalize. Along the way we note various structural properties of stabilizer states, with respect to their orthogonality relationships, which may be of independent interest.