  Let $\mathcal Q_D$ be the set of moduli points on Siegel's modular threefold
whose corresponding principally polarized abelian surfaces have quaternionic
multiplication by a maximal order $\mathcal O$ in an indefinite quaternion
algebra of discriminant $D$ over $\mathbb Q$ such that the Rosati involution
coincides with a positive involution of the form
$\alpha\mapsto\mu^{-1}\overline\alpha\mu$ on $\mathcal O$ for some
$\mu\in\mathcal O$ with $\mu^2+D=0$. In this paper, we first give a formula for
the number of irreducible components in $\mathcal Q_D$, strengthening an
earlier result of Rotger. Then for each irreducible component of genus $0$, we
determine its rational parameterization in terms of a Hauptmodul of the
associated Shimura curve.
