Recently, aspect sentiment understanding in conversational texts has received growing attention from researchers, leading to notable progress in the field. Unlike prior research, this study focuses on the performance of large language models (LLMs) in conversational aspect sentiment understanding tasks and highlights two primary challenges: aspect-coreference mapping and aspect-sentiment mapping. These challenges pose significant constraints on achieving accurate aspect sentiment understanding within conversational structures. To address these issues, the study defines a new task for LLMs in conversational aspect sentiment understanding, aiming to extract quadruples that encapsulate both aspect-coreference mapping and aspect-sentiment mapping relationships. A high-quality annotated dataset of conversational aspect sentiment quadruples has been created to evaluate the performance of LLMs on this task. To tackle the challenges of mapping relationships and mitigate the inherent hallucination issues of LLMs, this study introduces a multi-agent consistency reflection approach. This method involves the design of three sub-task agents to aid in capturing the complex mapping relationships within conversational structures. In addition, an enhanced consistency reflection is proposed, enabling the model to generate optimal results through multi-agent consensus, thus alleviating hallucination problems. Experimental results demonstrate that the proposed approach significantly outperforms state-of-the-art benchmarks. Furthermore, it exhibits superior capabilities in extracting aspect referential relationships and aspect sentiments, contributing to advancements in fine-grained sentiment understanding within conversational contexts using LLMs.