Quaternion matrix equations have widespread applications in signal processing, image processing, control systems, etc. By using the real representation, determinant, inverse matrix definition and properties of quaternion matrices, and based on the Kronecker product of quaternion matrices, a class of coupled Sylvester quaternion matrix equations is transformed into equivalent real linear equations. The Cramer's rule for the solution of the equations with a unique solution is studied. Finally, the Cramer's rule for the coupled Sylvester quaternion matrix equations is derived by using the isomorphic relationship between quaternions and their real representations. Through the analysis of the method, the theoretical validity and feasibility are proved. Numerical examples demonstrate that the proposed theoretical method has good computational performance and practicality.