The improvement in translating natural language to structured query language (SQL) can be attributed to the advancements in large language models (LLMs). Open-source LLMs, tailored for specific database dialects such as MySQL, have shown great performance. However, cloud service providers are looking for a unified database manager service (e.g., Cosmos DB from Azure, Amazon Aurora from AWS, Lindorm from AlibabaCloud) that can support multiple dialects. This requirement has led to the concept of multi-dialect query generation, which presents challenges to LLMs. These challenges include syntactic differences among dialects and imbalanced data distribution across multiple dialects. To tackle these challenges, we propose MoMQ, a novel Mixture-of-Experts-based multi-dialect query generation framework across both relational and non-relational databases. MoMQ employs a dialect expert group for each dialect and a multi-level routing strategy to handle dialect-specific knowledge, reducing interference during query generation. Additionally, a shared expert group is introduced to address data imbalance, facilitating the transfer of common knowledge from high-resource dialects to low-resource ones. Furthermore, we have developed a high-quality multi-dialect query generation benchmark that covers relational and non-relational databases such as MySQL, PostgreSQL, Cypher for Neo4j, and nGQL for NebulaGraph. Extensive experiments have shown that MoMQ performs effectively and robustly even in resource-imbalanced scenarios.