NoSQL databases have been widely adopted in big data analytics, geospatial applications, and healthcare services, due to their flexibility and scalability. However, querying NoSQL databases requires specialized technical expertise, creating a high barrier for users. While recent studies have explored text-to-NoSQL problem, they primarily focus on single-turn interactions, ignoring the conversational nature of real-world queries. To bridge this gap, we introduce the Conversational Text-to-NoSQL task, which generates NoSQL queries given a natural language question, a NoSQL database, and the dialogue history. To address this task, we propose Stage-MCTS, a framework that endows small language models (SLMs) with NoSQL-specific reasoning capabilities by formulating query generation as a search problem. The framework employs Monte Carlo Tree Search (MCTS) guided by a rule-based reward to produce stepwise reasoning data, followed by progressive supervised fine-tuning (SFT) and self-training strategies. We further construct CoNoSQL, a cross-domain dataset with over 2,000 dialogues and 150 databases, to support evaluation. Experiments demonstrate that our approach outperforms state-of-the-art large reasoning models, improving execution value match (EVM) accuracy by up to 7.93%.