Quantized low-rank matrix recovery estimates the original matrix from its entry-wise quantized measurements. Subspace clustering divides data points belonging to the union of subspaces (UoS) into the respective subspaces. Generalizing from both quantized matrix recovery and subspace clustering, this paper for the first time studies the problem of combined data recovery and subspace clustering based on the quantized measurements of data points following the UoS model. The recovery and clustering is achieved simultaneously by solving a nonconvex constrained maximum likelihood problem. The relative recovery error is proved to diminish to zero as the matrix size increases. A sparse alternative proximal algorithm with a convergence guarantee is proposed to solve the nonconvex problem. The proposed method is evaluated numerically on synthetic and extended Yale Face B datasets.