The rapid development of aspect-based sentiment analysis (ABSA) within recent decades shows great potential for real-world society. The current ABSA works, however, are mostly limited to the scenario of a single text piece, leaving the study in dialogue contexts unexplored. To bridge the gap between fine-grained sentiment analysis and conversational opinion mining, in this work, we introduce a novel task of conversational aspect-based sentiment quadruple analysis, namely DiaASQ, aiming to detect the quadruple of target-aspect-opinion-sentiment in a dialogue. We manually construct a large-scale high-quality DiaASQ dataset in both Chinese and English languages. We deliberately develop a neural model to benchmark the task, which advances in effectively performing end-to-end quadruple prediction, and manages to incorporate rich dialogue-specific and discourse feature representations for better cross-utterance quadruple extraction. We hope the new benchmark will spur more advancements in the sentiment analysis community.