This paper studies the relation among the number of spanning trees of intermediate graphs in a Galois cover, building on results for $(\mathbb{Z}/2\mathbb{Z})^m$-covers previously established by Hammer, Mattman, Sands, and Vallières. We generalize their results to arbitrary finite Galois covers. Using the Ihara zeta function and the Artin--Ihara $L$-function, we prove two formulas which are graph-theoretic analogues of Kuroda's formula and the Brauer--Kuroda relations in algebraic number theory. Furthermore, we prove that a spanning tree formula does not exist if the Galois group is cyclic.