Recent low-rank quaternion matrix completion (LRQMC) approaches have been extensively studied to recover missing data of color images. However, these methods need to frequently compute the quaternion singular value decompositions (QSVD) of the quaternion matrix, making them unsuitable for large-scale data. In this paper, we suggest an efficient LRQMC model based on the learnable transforms for color image recovery. The key idea is to project the large-scale quaternion matrix to a small-scale quaternion matrix via the semi-orthogonal transforms along each mode, which significantly reduces the computational cost of QSVD. We then apply a nonconvex approximation of rank (i.e., weighted Schatten p-norm) onto the small-scale quaternion matrix to achieve a better quaternion rank estimation. The alternating direction method of multipliers scheme is developed to solve the proposed model, and the weak convergence property of the algorithm is discussed. Experimental results on color images demonstrate that our method is considerably faster than state-of-art approaches while achieving comparative recovery performance.