The faithful quaternion character has Schur index two

In the usual complex model, $i$ and $j$ have eigenvalues $i,-i$, so their traces, and those of their negatives, are $0$; $-1$ acts as $-I_2$. Hence the displayed character is rational-valued.

Let ${\mathbb{H}}_{\mathbb{Q}}=\mathbb{Q}+\mathbb{Q}i+\mathbb{Q}j+\mathbb{Q}k$ and let $Q_8$ act on it by left multiplication. The norm $q\bar{q}=a^2+b^2+c^2+d^2$ is nonzero for every nonzero rational quaternion, so ${\mathbb{H}}_{\mathbb{Q}}$ is a division algebra. A $Q_8$-stable rational subspace is therefore a left ideal (the elements of $Q_8$ span ${\mathbb{H}}_{\mathbb{Q}}$), and this four-dimensional rational representation is irreducible.

The trace of left multiplication is $4$ at $1$, $-4$ at $-1$, and $0$ at the other six elements. Thus the complexification of the irreducible rational representation in step 1.2 has character $2\chi$ (equivalently, ${\mathbb{H}}_{\mathbb{Q}}{\otimes }_{\mathbb{Q}}\mathbb{C}\cong M_2(\mathbb{C})$ is two copies of the natural module under left multiplication). This explicit multiplicity computation uses no choice principle. Under AC, [L1] identifies this irreducible rational module with the attached one, so its multiplicity is $m_{\mathbb{Q}}(\chi )=2$.