The generalized singular value decomposition (GSVD) is an important tool in matrix computations and has been widely applied in areas such as regularization and signal processing. However, as the scale of data sets increases, classical methods face significant challenges in terms of computational time and storage complexity when applied to large-scale matrix pairs. To address this issue, a randomized GSVD method based on low-rank approximation is proposed. The method first uses random projection to capture the dominant subspaces of the matrices and constructs low-rank approximations via orthogonal factorization, thereby compressing the original matrix pairs. The GSVD of the compressed small matrix pair is then computed in the compressed space. The structural properties and computational complexity of the algorithm are analyzed, and the effectiveness of the method is established. Numerical experiments demonstrate that, in the low-rank setting, the proposed method maintains numerical stability while achieving a three- to four-fold speedup compared with the built-in GSVD function in Matlab, along with high accuracy. Furthermore, in the regularized solution of ill-posed problems, the method is shown to effectively improve the quality of approximate solutions.