ag.algebraic geometry - Understanding different Q-models of a curve over C

Yes.

Roughly, the idea is that once your base object $X/B$ is fixed then for any object $Y$ you can consider the sheaf $Iso(X,Y)$ of isomorphisms from $X$ to $Y$. This has an action of the sheaf $Aut(X)$ and on any open cover $U$ where $Y|_U cong X|_U$, making a choice of such an isomorphism gives an isomorphism of $Aut(X)(U)$-sets $Iso(X,Y)(U) cong Aut(X)(U)$.

The difference from the standard argument in this case is that you need to use the etale topology instead of the Zariski topology. The etale covers of $mathbb{Q}$ are finite field extensions $F$ and so the statement that $Y$ is "locally" equivalent to $X$ means that $Y_F cong X_F$ for a large enough field extension $F$.