Firstly, since you are interested in the field of definition of the surfaces, you should work with the moduli stack, rather than the coarse moduli space. A $k$-rational point on the moduli stack corresponds to a surface defined over $k$, but the same is not true for the coarse moduli space (this is the issue that ``Field of definition $\neq$ Field of moduli'' in general for coarse moduli spaces; though this issue only occurs for surfaces with non-trivial automorphism group).
So let $\mathcal{X}$ be moduli stack of interest. The functor of points is as follows: for a $\mathbb{Q}$-scheme $S$, we have that $\mathcal{X}(S)$ is the set of all smooth proper morphisms $Y \to S$ such that the relative canonical bundle $\omega_{Y/S}$ is relatively ample.
You seem to be also interested in singular surfaces, so you should adapt the definition as required; but this will correspond to some compactification of this stack. As you are interested in density of rational points, one can just work with the open subset corresponding to smooth surfaces. Moreover, if you want a stack of finite type, you should restrict the Hodge numbers or Hilbert polynomial, as you have done in the question.
Now note that the definition makes sense for any $\mathbb{Q}$-scheme $S$, so this means that $\mathcal{X}$ is defined over $\mathbb{Q}$. In fact, you can easily modify the functor of points definition for an arbitrary scheme $S$, which means that one can define a moduli stack over $\mathbb{Z}$, just as Deligne-Mumford defined the moduli stack over curves of given genus over $\mathbb{Z}$.
Next, in general, unless there is a good reason otherwise, one expects that moduli spaces are of general type. Therefore conjectures of Lang-Vojta predict that the $\mathbb{Q}$-rational points are not Zariski dense. Hence in general there is no reason to expect to be able to approximate a surface over $\mathbb{C}$ by a surface over $\mathbb{Q}$.
As for $\bar{\mathbb{Q}}$-points, as explained in the comments, for a non-empty finite type $\bar{\mathbb{Q}}$ scheme, the $\bar{\mathbb{Q}}$-points are Zariski dense.