Cough is a key symptom reflecting respiratory health, with its frequency and pattern providing valuable insights into disease progression and clinical management. Objective and reliable cough detection systems are therefore of broad significance for healthcare and remote monitoring. However, existing algorithms often struggle to jointly model spatial and temporal information, limiting their robustness in real-world applications. To address this issue, we propose a cough recognition framework based on frequency-modulated continuous-wave (FMCW) radar, integrating a deep convolutional neural network (CNN) with a Self-Attention mechanism. The CNN extracts spatial features from range-Doppler maps, while Self-Attention captures temporal dependencies, and effective data augmentation strategies enhance generalization by simulating position variations and masking local dependencies. To rigorously evaluate practicality, we collected a large-scale radar dataset covering diverse positions, orientations, and activities. Experimental results demonstrate that, under subject-independent five-fold cross-validation, the proposed model achieved a mean F1-score of $0.974\pm 0.016$ and an accuracy of $99.05\pm 0.55$ %, further supported by high precision of $98.77\pm 1.05$ %, recall of $96.07\pm 2.16$ %, and specificity of $99.73\pm 0.23$ %. These results confirm that our method is not only robust in realistic scenarios but also provides a practical pathway toward continuous, non-invasive, and privacy-preserving respiratory health monitoring in both clinical and telehealth applications.