Transformer Hawkes process models have shown to be successful in modeling event sequence data. However, most of the existing training methods rely on maximizing the likelihood of event sequences, which involves calculating some intractable integral. Moreover, the existing methods fail to provide uncertainty quantification for model predictions, e.g., confidence interval for the predicted event's arrival time. To address these issues, we propose SMURF-THP, a score-based method for learning Transformer Hawkes process and quantifying prediction uncertainty. Specifically, SMURF-THP learns the score function of the event's arrival time based on a score-matching objective that avoids the intractable computation. With such a learnt score function, we can sample arrival time of events from the predictive distribution. This naturally allows for the quantification of uncertainty by computing confidence intervals over the generated samples. We conduct extensive experiments in both event type prediction and uncertainty quantification on time of arrival. In all the experiments, SMURF-THP outperforms existing likelihood-based methods in confidence calibration while exhibiting comparable prediction accuracy.