Quaternion linear systems have wide applications in both the control theory and engineering. Quaternion matrix equations are studied by means of semi-tensor product of matrices. A real vector representation of quaternion matrix is presented and its properties are studied. Combining the real vector representation and the semi-tensor product of matrices, the equivalent conditions for existence of Hermitian solution and the expressions of the minimal norm Hermitian solution of quaternion matrix equation $A^HXA=B$ are proposed, and the corresponding algorithm is given. The effectiveness of the real vector representation method is verified by numerical experiments.