A conversational music retrieval system can help users discover music that matches their preferences through dialogue. To achieve this, a conversational music retrieval system should seamlessly engage in multi-turn conversation by 1) understanding user queries and 2) responding with natural language and retrieved music. A straightforward solution would be a data-driven approach utilizing such conversation logs. However, few datasets are available for the research and are limited in terms of volume and quality. In this paper, we present a data generation framework for rich music discovery dialogue using a large language model (LLM) and user intents, system actions, and musical attributes. This is done by i) dialogue intent analysis using grounded theory, ii) generating attribute sequences via cascading database filtering, and iii) generating utterances using large language models. By applying this framework to the Million Song dataset, we create LP-MusicDialog, a Large Language Model based Pseudo Music Dialogue dataset, containing over 288k music conversations using more than 319k music items. Our evaluation shows that the synthetic dataset is competitive with an existing, small human dialogue dataset in terms of dialogue consistency, item relevance, and naturalness. Furthermore, using the dataset, we train a conversational music retrieval model and show promising results.