Vision-language alignment using chest X-rays and radiology reports has emerged as an advanced paradigm for zero-shot classification and grounding of chest X-ray findings. However, standard contrastive learning typically treats radiographs and reports from different patients simply as negative pairs. This assumption introduces noisy negatives, as different patients frequently exhibit similar findings. Such noisy negatives cause semantic ambiguity and degrade performance in zero-shot understanding tasks. To address this challenge, we propose CoNNS, a concept-guided noisy-negative suppression framework. To support the negative suppression mechanism, unlike previous methods that use raw reports or templatized texts, we construct a hierarchical concept ontology using large language models. The ontology structures 41 key clinical concepts by explicitly modeling presence, attributes (location and characteristics), and texts (evidential segment and presence statement). Leveraging this ontology, we implement a cross-patient pair relabeling strategy comprising three steps: (1) Fine-Grained Breakdown to categorize pairs based on finding presence; (2) Noisy Negative Filtering to resolve semantic conflicts by removing false negatives; and (3) Hard Negative Mining to identify subtle attribute discrepancies using a lightweight language model. Finally, we propose a Concept-Aware NCE loss to align visual features with text while suppressing the identified noisy negatives. Extensive experiments across multi-granularity zero-shot grounding tasks and five zero-shot classification datasets validate that CoNNS outperforms existing state-of-the-art models. The code is available at https://github.com/DopamineLcy/conns.