Recently I was wondering about generalizations of Beyli's theorem to higher dimensions and did some googling. As this issue is only discussed briefly in David Roberts' comment, I thought I contribute what references I found hoping someone might find it useful:

There is one direction of research which looks for actions of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on algebraic objects other than curves. A general criterion for an algebraic variety to be defined over a number field has been provided in this paper of Gabino González-Diez. In the special case of surfaces, there was a question of Catanese on the Galois action on moduli surfaces. This question has been answered in papers of Easton and Vakil (published in International Math. Res. Notices 20, 2007) and of Bauer, Catanese and Grunewald.

There is an interesting survey of Goldring (published in the Serge Lang memorial proceedings "Number theory, analysis and geometry"), which also discusses higher-dimensional generalizations of Belyi's theorem. Apparently one way of generalizing Belyi's theorem is

Braungardt's question: Is every connected quasi-projective variety $X$ that is defined over $\overline{\mathbb{Q}}$ birational to a finite étale cover of some moduli space of curves $\mathcal{M}_{g,n}$?

It was formulated in the paper: V. Braungardt: Covers of moduli surfaces. Compositio Math. 140 (2004) 1033-1036. In this paper, there are also some partial results on this question. There is also a paper of Paranjape on realization of surfaces defined over $\overline{\mathbb{Q}}$ as branched covers of $\mathbb{P}^2$.