Upmixing consists in extracting audio objects out of their downmix, given some parameters computed beforehand at a coding stage. It is an important task in audio processing with many applications in the entertainment industry. One particularly successful approach for this purpose is to compress the audio objects through nonnegative matrix factorization (NMF) parameters at the coder, to be used for separating the downmix at the decoder. In this paper, we focus on such NMF methods for audio compression, which operate at very low parameter bitrates. In existing methods, parameter estimation and quantization are conducted independently. Here, we propose two extensions: first, we jointly estimate and quantize the parameters at the coder to ensure good reconstruction at the decoder. Second, we propose a parameter refinement method operated at the decoder, that benefits from priors induced by quantization to yield better performance. We show that our contributions outperform existing baseline methods.