<p>This paper presents a novel randomized Quaternion Singular Value Decomposition (QSVD) algorithm with orthogonal constraints, specifically tailored for non-standard inner product. In fact, the traditional QSVD method often suffers from huge computational cost and the redundant information. To address these issues, we employ a sketch matrix and an oblique projector to effectively reduce the dimensionality of the original quaternion matrix while preserving its essential properties. By extending the algorithm to Hermitian quaternion matrices with a two-sided oblique projector, we facilitate the application of our method to Quaternion Generalized eigenvalue Decomposition (QGED), significantly broadening its utility and effectiveness. Moreover, the theoretical analysis establishes error bounds under variant contexts, ensuring the robustness of our algorithms. Finally, numerical experiments further demonstrate that our algorithms achieve high approximation accuracy within an acceptable error range.</p>