The main focus of this study is on the relative controllability of linear and semilinear quaternion differential equations with delay. The Gram criterion and the rank criterion for the relative controllability of linear quaternion differential equations (LQDEs) with pure delay are presented by the delayed quaternion Gram matrix and the maximal right linearly independent group, respectively. Furthermore, a pure delay-based minimal norm control of LQDEs is achieved. For LQDEs with delay, we provide a necessary and sufficient condition for relative controllability and eliminate the requirement for permutation matrices. The relative controllability requirement for semi-LQDEs (SLQDEs) is then established using the Krasnoselskii's fixed point theorem. Some examples are given to illustrate our theoretical results.