We consider the Ihara zeta function $ζ(u,X//G)$ and Artin-Ihara $L$-function of the quotient graph of groups $X//G$, where $G$ is a group acting on a finite graph $X$ with trivial edge stabilizers. We determine the relationship between the primes of $X$ and $X//G$ and show that $X\to X//G$ can be naturally viewed as an unramified Galois covering of graphs of groups. We show that the $L$-function of $X//G$ evaluated at the regular representation is equal to $ζ(u,X)$ and that $ζ(u,X//G)$ divides $ζ(u,X)$. We derive two-term and three-term determinant formulas for the zeta and $L$-functions, and compute several examples of $L$-functions of edge-free quotients of the tetrahedron graph $K_4$.