Some necessary and sucient conditions for the existence of the -skew-Hermitian solution quaternion matrix equations the system of matrix equations with -skew-hermicity, A1X = C1; XB1 = C2; A2Y = C3; YB2 = C4; X = X; Y = Y; A3XA 3 + B3YB 3 = C5; are established in this paper by using rank equalities of the coecient matrices. The general solutions to the system and its special cases are provided when they are consistent. Within the framework of the theory of noncommutative row-column determinants, we also give determinantal representation formulas of finding their exact solutions that are analogs of Cramer’s rule. A numerical example is also given to demonstrate the main results.