We study a quantum version of the SU(2) Hopf fibration S7 → S4 and its associated twistor geometry. Our quantum sphere Sq7 arises as the unit sphere inside a q-deformed quaternion space Hq2. The resulting four-sphere Sq4 is a quantum analogue of the quaternionic projective space HP1. The quantum fibration is endowed with com- patible non-universal differential calculi. By investigating the quantum symmetries of the fibration, we obtain the geometry of the corresponding twistor space CPq3 and use it to study a system of anti-self-duality equations on Sq4, for which we find an ‘instanton’ solution coming from the natural projection defining the tautological bundle over Sq4
