The $p$-hub center problem is a fundamental model for the strategic design of hub location. It aims at constructing $p$ fully interconnected hubs and links from nodes to hubs so that the longest path between any two nodes is minimized. Existing literature on the $p$-hub center problem under uncertainty often assumes a joint distribution of travel times, which is difficult (if not impossible) to elicit precisely. In this paper, we bridge the gap by investigating two distributionally robust chance constrained models, which cover the existing stochastic one with independent normal distribution and the sample average approximation approach as a special case, respectively. We derive deterministic reformulations as a mixed-integer program wherein a large number of constraints can be dynamically added via a constraint generation approach to accelerate computation. Extensive numerical experiments demonstrate the encouraging out-of-sample performance of our proposed models as well as the effectiveness of the constraint generation approach.