Image quality transfer (IQT) aims to enhance clinical images of relatively low quality by learning and propagating high-quality structural information from expensive or rare data sets. However, the original framework gives no indication of confidence in its output, which is a significant barrier to adoption in clinical practice and downstream processing. In this work, we present a general Bayesian extension of IQT which enables efficient and accurate quantification of uncertainty, providing users with an essential prediction of the accuracy of enhanced images. We demonstrate the efficacy of the uncertainty quantification through super-resolution of diffusion tensor images of healthy and pathological brains. In addition, the new method displays improved performance over the original IQT and standard interpolation techniques in both reconstruction accuracy and robustness to anomalies in input images.