Edge applications, such as collaborative robotics and spacecraft rendezvous, demand efficient 6D object pose estimation on resource-constrained embedded platforms. Existing 6D object pose estimation networks are often too large for such deployments, necessitating compression while maintaining reliable performance. To address this challenge, we introduce Modular Quantization-Aware Training (MQAT), an adaptive and mixed-precision quantization-aware training strategy that exploits the modular structure of modern 6D object pose estimation architectures. MQAT guides a systematic gradated modular quantization sequence and determines module-specific bit precisions, leading to quantized models that outperform those produced by state-of-the-art uniform and mixed-precision quantization techniques. Our experiments showcase the generality of MQAT across datasets, architectures, and quantization algorithms. Additionally, we observe that MQAT quantized models can achieve an accuracy boost (>7% ADI-0.1d) over the baseline full-precision network while reducing model size by a factor of 4x or more.