Phylodynamics is used to estimate epidemic dynamics from phylogenetic trees or genomic sequences of pathogens, but the likelihood calculations needed can be challenging for complex models. We present a neural Bayes estimator (NBE) for key epidemic quantities: the reproduction number, prevalence, and cumulative infections through time. By performing quantile regression over tree space, the NBE allows us to estimate posterior medians and credible intervals directly from a reconstructed tree. Our approach uses a recursive neural network as a tree embedding network with a prediction network conditioned on time and quantile level to generate the estimates. In simulation studies, the NBE achieves good predictive performance, with conservative uncertainty estimates. Compared with a BEAST2 fixed-tree analysis, the NBE gives less biased estimates of time-varying reproduction numbers in our test setting. Under a misspecified sampling model, the NBE performance degrades (as expected) but remains reasonable, and fine-tuning a pre-trained model yields estimates comparable to those from a model trained from scratch, at substantially lower computational cost.