Large language models (LLMs) have emerged as strong contenders in machine translation.Yet, they still struggle to adequately handle discourse phenomena, such as pronoun resolution and lexical cohesion at the document level. In this study, we thoroughly investigate the discourse phenomena performance of LLMs in context-aware translation. We demonstrate that discourse knowledge is encoded within LLMs and propose the use of quality-aware decoding (QAD) to effectively extract this knowledge, showcasing its superiority over other decoding approaches through comprehensive analysis. Furthermore, we illustrate that QAD enhances the semantic richness of translations and aligns them more closely with human preferences.