In this article, we show that sets of three-qubit quantum observables obtained by considering both the classical and skew embeddings of the split Cayley hexagon of order two into the binary symplectic polar space of rank three can be used to detect quantum state-independent contextuality. This reveals a fundamental connection between these two appealing structures and some fundamental tools in quantum mechanics and quantum computation. More precisely, we prove that the complement of a classically embedded hexagon does not provide a Mermin–Peres-like proof of the Kochen–Specker theorem whereas that of a skewly-embedded one does.