Object counting in images has been studied extensively, in particular using deep network models recently. The existing counting models typically output the point estimates of the object counts in given images. However, none of these can provide reliable uncertainty quantification of the derived count estimates, which is critical for consequent decision making when adopting these counting models in real-world applications. In this paper, we propose a novel deep counting model in a Bayesian framework. With the designed Bayesian attention module and Bayesian counting loss function, our deep Bayesian counting model not only improves the accuracy of count estimates with varying object and background appearance; but also enables their uncertainty quantification. We specifically focus on plant counting, which plays important roles in AI-augmented agriculture, for example crop yield estimates and farm management. Our ablation studies and experiments with the real-world agriculture data in the Global Wheat dataset have demonstrated that our deep Bayesian counting model obtains high count estimation accuracy as well as reliable uncertainty quantification. In addition, with the integrated Bayesian attention modules, it may help improve the interpretability of the derived count estimates, especially when the distribution of the interested plants in images is heterogeneous.