Medical imaging plays a pivotal role in diagnosis and treatment in clinical practice. Inspired by the significant progress in automatic image captioning, various deep learning (DL)-based architectures have been proposed for generating radiology reports for medical images. However, model uncertainty (i.e., model reliability/confidence on report generation) is still an under-explored problem. In this paper, we propose a novel method to explicitly quantify both the visual uncertainty and the textual uncertainty for the task of radiology report generation. Such multi-modal uncertainties can sufficiently capture the model confidence scores at both the report-level and the sentence-level, and thus they are further leveraged to weight the losses for achieving more comprehensive model optimization. Our experimental results have demonstrated that our proposed method for model uncertainty characterization and estimation can provide more reliable confidence scores for radiology report generation, and our proposed uncertainty-weighted losses can achieve more comprehensive model optimization and result in state-of-the-art performance on a public radiology report dataset.