Integrating Large Language Models (LLMs) with existing Knowledge Graph (KG) databases presents a promising avenue for enhancing LLMs' efficacy and mitigating their "hallucinations". Given that most KGs reside in graph databases accessible solely through specialized query languages (e.g., Cypher), there exists a critical need to bridge the divide between LLMs and KG databases by automating the translation of natural language into Cypher queries (commonly termed the "Text2Cypher" task). Prior efforts tried to bolster LLMs' proficiency in Cypher generation through Supervised Fine-Tuning. However, these explorations are hindered by the lack of annotated datasets of Query-Cypher pairs, resulting from the labor-intensive and domain-specific nature of annotating such datasets. In this study, we propose SyntheT2C, a methodology for constructing a synthetic Query-Cypher pair dataset, comprising two distinct pipelines: (1) LLM-based prompting and (2) template-filling. SyntheT2C facilitates the generation of extensive Query-Cypher pairs with values sampled from an underlying Neo4j graph database. Subsequently, SyntheT2C is applied to two medical databases, culminating in the creation of a synthetic dataset, MedT2C. Comprehensive experiments demonstrate that the MedT2C dataset effectively enhances the performance of backbone LLMs on the Text2Cypher task. Both the SyntheT2C codebase and the MedT2C dataset will be released soon.