Uncertainty quantification in medical image registration is crucial for doctors to evaluate risk in real-world clinical applications. Recently， deep unsupervised learning-based medical image registration models have shown certain effects， but there is a lack of methods to estimate appearance uncertainty during registration， which will affect registration accuracy and trustworthiness. In addition， in real-time application scenarios， medical image registration models need to be highly accurate and fast in inference at the same time， which is difficult to be achieved by the existing models. To address these issues， an uncertainty-aware unsupervised medical image registration model based on Evidential Deep Learning （EDL） — EvidentialMorph was proposed to apply EDL， an uncertainty quantification approach without additional computational cost， to unsupervised medical image registration. Firstly， the Deformation Vector Field （DVF） was learnt and obtained through a registration backbone network module with a U-net architecture. Then， the Normal-Inverse Gamma （NIG） distribution of the registered image was learnt and obtained through an improved Spatial Transformer Network （STN） module — evidential STN module， thereby calculating the registered image and its appearance uncertainty directly. Experiments were carried out on Hippocampus， LPBA40， and IBSR18 Magnetic Resonance Imaging （MRI） datasets. The results show that in registration accuracy， EvidentialMorph improves the Dice Similarity Coefficient （DSC） and Normalized Cross-Correlation （NCC） coefficient by 3.31% and 2.75% at most， respectively， over CLMorph model； and in inference time， EvidentialMorph reduces 85 ms. The above results verify that EvidentialMorph can obtain effective uncertainty quantification quickly and improves registration accuracy， offering potential for real-time medical image registration scenarios and improving registration effects.