In this paper, we propose QUITE, a novel approach that leverages a large-scale language model (LLM) to overcome the limitations of existing rule-based SQL query rewriting methods. Existing methods rely on predefined rules, which limits the diversity of query patterns and rewriting strategies and can cause performance degradation. QUITE is a learning-free feedback-aware system based on LLM agents. It solves the hallucination problem of LLM, supports a wider range of query patterns and rewriting strategies, and improves performance through a multi-agent framework using a finite state machine (FSM), a rewriting middleware, and a hint injection technique. Experimental results show that QUITE reduces query execution time by up to 35.8% compared to state-of-the-art methods, and rewrites 24.1% more queries than existing methods.