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Sparse signals, encountered in many wireless and signal acquisition applications, can be acquired via compressed sensing (CS) to reduce computations and transmissions, crucial for resource-limited devices, e.g., wireless sensors. Since the information signals are often continuous-valued, digital communication of compressive measurements requires quantization. In such a quantized compressed sensing (QCS) context, we address remote acquisition of a sparse source through vector quantized noisy compressive measurements. We propose a deep encoder-decoder architecture, consisting of an encoder deep neural network (DNN), a quantizer, and a decoder DNN, that realizes low-complexity vector quantization aiming at minimizing the mean-square error of the signal reconstruction for a given quantization rate. We devise a supervised learning method using stochastic gradient descent and backpropagation to train the system blocks. Strategies to overcome the vanishing gradient problem are proposed. Simulation results show that the proposed non-iterative DNN-based QCS method achieves higher rate-distortion performance with lower algorithm complexity as compared to standard QCS methods, conducive to delay-sensitive applications with large-scale signals.