Just in case anyone else is still thinking about this question...
The answer is the following. Either:
The eigenvalues of $G_n$ are all of the form $\zeta + \zeta^{-1}$ for roots of unity $\zeta$, and the graphs $G_n$ are subgraphs of the (extended) Dynkin diagrams $\widetilde{A}_n$ or $\widetilde{D}_n$.
For sufficiently large $n$, the largest eigenvalue $\lambda$ is greater than two, and $\mathbf{Q}(\lambda^2)$ is not abelian.
The proof is effective, but a little long to post here. The main ingredients are some basic facts about Weil height, some ideas due to Cassels, and an amplification step using Chebyshev polynomials.