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Rate-Distortion Optimized Quantization (RDOQ) has played an important role in the coding performance of recent video compression standards such as H.264/AVC, H.265/HEVC, VP9 and AV1. This scheme yields significant reductions in bit-rate at the expense of relatively small increases in distortion. Typically, RDOQ algorithms are prohibitively expensive to implement on real-time hardware encoders due to their sequential nature and their need to frequently obtain entropy coding costs. This work addresses this limitation using a neural network-based approach, which learns to trade-off rate and distortion during offline supervised training. As these networks are based solely on standard arithmetic operations that can be executed on existing neural network hardware, no additional area-on-chip needs to be reserved for dedicated RDOQ circuitry. We train two classes of neural networks, a fully-convolutional network and an auto-regressive network, and evaluate each as a post-quantization step designed to refine cheap quantization schemes such as scalar quantization (SQ). Both network architectures are designed to have a low computational overhead. After training they are integrated into the HM 16.20 implementation of HEVC, and their video coding performance is evaluated on a subset of the H.266/VVC SDR common test sequences. Comparisons are made to RDOQ and SQ implementations in HM 16.20. Our method achieves 1.64% BD-rate savings on luminosity compared to the HM SQ anchor, and on average reaches 45% of the performance of the iterative HM RDOQ algorithm.