The rise of IoT devices has prompted the demand for deploying machine learning at-the-edge with real-time, efficient, and secure data processing. In this context, implementing machine learning (ML) models with real-valued weight parameters can prove to be impractical particularly for large models, and there is a need to train models with quantized discrete weights. At the same time, these low-dimensional models also need to preserve privacy of the underlying dataset. In this work, we present RQP-SGD, a new approach for privacy-preserving quantization to train machine learning models for low-memory ML-at-the-edge. This approach combines differentially private stochastic gradient descent (DP-SGD) with randomized quantization, providing a measurable privacy guarantee in machine learning. In particular, we study the utility convergence of implementing RQP-SGD on ML tasks with convex objectives and quantization constraints and demonstrate its efficacy over deterministic quantization. Through experiments conducted on two datasets, we show the practical effectiveness of RQP-SGD.