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This paper focuses on Dialogue Aspect-based Sentiment Quadruple (DiaASQ) analysis, aiming to extract structured quadruples from multi-turn conversations. Applying Large Language Models (LLMs) for this specific task presents two primary challenges: the accurate extraction of multiple elements and the understanding of complex dialogue reply structure. To tackle these issues, we propose a novel LLM-based multi-task approach, named Task-aware Contrastive Mixture of Experts (TaCoMoE), to tackle the DiaASQ task by integrating expert-level contrastive loss within task-oriented mixture of experts layer. TaCoMoE minimizes the distance between the representations of the same expert in the semantic space while maximizing the distance between the representations of different experts to efficiently learn representations of different task samples. Additionally, we design a Graph-Centric Dialogue Structuring strategy for representing dialogue reply structure and perform non-opinion utterances detection to enhance the performance of quadruple extraction. Extensive experiments are conducted on the DiaASQ dataset, demonstrating that our method significantly outperforms existing parameter-efficient fine-tuning techniques in terms of both accuracy and computational efficiency. The code is available at https://github.com/he2720/TaCoMoE.