Visual place recognition (VPR) aiming at predicting the location of an image based solely on its visual features is a fundamental task in robotics and autonomous systems. Domain variation remains one of the main challenges in VPR and is relatively unexplored. Existing VPR models attempt to achieve domain agnosticism either by training on large-scale datasets that inherently contain some domain variations, or by being specifically adapted to particular target domains. In practice, the former lacks explicit domain supervision, while the latter generalizes poorly to unseen domain shifts. This paper proposes a novel query-based domain-agnostic VPR model called QdaVPR. First, a dual-level adversarial learning framework is designed to encourage domain invariance for both the query features forming the global descriptor and the image features from which these query features are derived. Then, a triplet supervision based on query combinations is designed to enhance the discriminative power of the global descriptors. To support the learning process, we augment a large-scale VPR dataset using style transfer methods, generating various synthetic domains with corresponding domain labels as auxiliary supervision. Extensive experiments show that QdaVPR achieves state-of-the-art performance on multiple VPR benchmarks with significant domain variations. Specifically, it attains the best Recall@1 and Recall@10 on nearly all test scenarios: 93.5%/98.6% on Nordland (seasonal changes), 97.5%/99.0% on Tokyo24/7 (day-night transitions), and the highest Recall@1 across almost all weather conditions on the SVOX dataset. Our code will be released at https://github.com/shuimushan/QdaVPR.