The uniformly accelerated Unruh-DeWitt detector serves as a fundamental model in relativistic quantum metrology. While previous studies have mainly concentrated on single-parameter estimation via quantum Cramér–Rao bound, the multi-parameter case remains significantly underexplored. In this paper, we investigate the multiparameter estimation for a uniformly accelerated Unruh-DeWitt detector coupled to a vacuum scalar field in both bounded and unbounded Minkowski vacuum. Our analysis reveals that quantum Cramér–Rao bound fails to provide a tight error bound for the two-parameter estimation involving the initial phase and weight parameters. For this reason, we numerically compute two tighter error bounds, Holevo Cramér–Rao bound and Nagaoka bound, based on a semidefinite program. Notably, our results demonstrate that the Nagaoka bound yields the tightest error bound among all the bounds considered and is attained by the four-outcome positive-operator-valued measures constructed, consistent with the general hierarchy of multiparameter quantum estimation. In the case with a boundary, we observe the introduction of boundary systematically reduces the values of both Holevo Cramér–Rao bound and Nagaoka bound, indicating an improvement on the attainable estimation precision. These results offer valuable insights on and practical guidance for advancing multiparameter estimation in relativistic context.