Do elements of the absolute Galois group $G=\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ of $\mathbf{Q}$ induce automorphisms of $\mathbf{C}$ if we extend the morphisms trivially to the rest of $\mathbf{C}$? Here we consider an element of $G$ as an automorphism of $\overline{\mathbf{Q}}\subset \mathbf{C}$ (and we fix this embedding).
Does this define an action on the moduli space of compact connected Riemann surfaces of genus $g$ by pulling back a Riemann surface via this action?
That is, let $\sigma$ be in $G$. Let $X$ be compact connected Riemann surface. Then we can let $\sigma$ act on $X$ algebraically by pulling back $X/\mathrm{Spec} \ \mathbf{C}$ along $\sigma$.
It is clear that $G$ acts on the $\overline{\mathbf{Q}}$-points of the moduli space $M_g$.