<p>Sylvester matrix equations serve as a powerful tool for representing mathematical models across various fields and have numerous applications. This paper investigates a class of generalized coupled Sylvester-type matrix equations in quaternion algebras. We introduce quaternion biconjugate gradient (QBiCG) method and quaternion quasi-minimum residual (QQMR) method for solving the generalized quaternion matrix equations. Both methods are based on the quaternion Lanczos biorthogonal procedure and utilize the conjugate linear operator derived from the real inner product defined in quaternion algebra. In the experimental section, we present a novel framework for the encryption and decryption of color images based on quaternion matrix equations. The numerical results demonstrate the efficacy of both the QBiCG and QQMR methods in solving quaternion Sylvester-type matrix equations.</p>