In the era of big data and rapidly evolving information systems, efficient and accurate data retrieval has become increasingly crucial. Neural graph databases (NGDBs) have emerged as a powerful paradigm that combines the strengths of graph databases (graph DBs) and neural networks to enable efficient storage, retrieval, and analysis of graph-structured data. The usage of neural embedding storage and complex neural logical query answering provides NGDBs with generalization ability. When the graph is incomplete, by extracting latent patterns and representations, neural graph databases can fill gaps in the graph structure, revealing hidden relationships and enabling accurate query answering. Nevertheless, this capability comes with inherent trade-offs, as it introduces additional privacy risks to the database. Malicious attackers can infer more sensitive information in the database using well-designed combinatorial queries, such as by comparing the answer sets of where Turing Award winners born before 1950 and after 1940 lived, the living places of Turing Award winner Hinton are probably exposed, although the living places may have been deleted in the training due to the privacy concerns. In this work, inspired by the privacy protection in graph embeddings, we propose a privacy-preserving neural graph database (P-NGDB) to alleviate the risks of privacy leakage in NGDBs. We introduce adversarial training techniques in the training stage to force the NGDBs to generate indistinguishable answers when queried with private information, enhancing the difficulty of inferring sensitive information through combinations of multiple innocuous queries. Extensive experiment results on three datasets show that P-NGDB can effectively protect private information in the graph database while delivering high-quality public answers responses to queries.