Knowledge graphs (KGs) have attracted increasing attention due to their fundamental role in many tasks. The quality of these KGs varies significantly, making their evaluation more and more critical. Existing methods for KG quality evaluation typically focus on either introducing new quality metrics across different dimensions or assessing performance during the KG construction process. However, these approaches are highly dependent on the raw data within the KGs, which may compromise internal information privacy. In practical applications, raw data needs to be strictly protected due to concerns such as commercial confidentiality. To address these challenges, we propose a novel knowledge graph quality evaluation framework under incomplete information (QEII). This framework reframes the evaluation task as an adversarial Q&A game between two KGs, where the winner is regarded as the higher-quality KG. Notably, our framework ensures privacy by preventing the exposure of raw data during the evaluation process. Experimental results on four KG pairs demonstrate that, compared to baseline methods, our framework effectively evaluates quality without disclosing private information.