Conversational Aspect-Based Sentiment Quadruple Analysis (DiaASQ), a newly emerged task, aims to extract target-aspect-opinion-sentiment quadruples from dialogues. Although demonstrating promising performance, existing methods struggle to effectively integrate diverse token-level information, and face challenges in modeling cross-utterance relationships, which hinders progress in this task. To address these limitations, the paper introduces the Task-Specific Token-Level Representations and Cross-Utterance Dependencies network (TTRCD), a novel network that not only enables effective integration of diverse information at the token-level but also models conversational thread-specific dependencies. On the one hand, we leverage the Context–Syntax Weaver (CSW) to obtain task-specific token-level representations by an adaptive fusion of contextual and syntactic structure information, which enriches the foundation for quadruple extraction. On the other hand, the Conversational Thread Weaver (CTW) is proposed to meticulously model the interplay of utterance-level dependencies within the thread, leveraging both direct and indirect interactions to construct a comprehensive understanding of the dialogue. Extensive experimental results on the DiaASQ benchmark confirm that the TTRCD network outperforms state-of-the-art methods, achieving significant improvements in quadruple extraction accuracy.