I would be most grateful if someone would check my logic below... Many thanks!
Let $G$ be a finite group and let $\bar{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$ inside $\mathbb{C}.$
Standard results from Representation Theory tells us the following:
(1) the number of irreducible $\mathbb{C}$-valued characters of $G$ is equal to the number of conj classes of $G.$
(2) an irreducible representation over a field $F$ remains irreducible over any extension field of $F$ if and only if it remains irreducible over the algebraic closure of $F.$
But (1) still goes through if we replace $\mathbb{C}$ by $\bar{\mathbb{Q}}$ (right?).
Therefore surely we have that
$$\{\text{irreducible $\mathbb{C}$-valued characters of $G$}\} = \{\text{irreducible $\bar{\mathbb{Q}}$-valued characters of $G$}\}?$$