  Principally polarized abelian surfaces with prescribed real multiplication
(RM) are parametrized by certain Hilbert modular surfaces. Thus rational genus
2 curves correspond to rational points on the Hilbert modular surfaces via
their Jacobians, but the converse is not true. We give a simple generic
description of which rational moduli points correspond to rational curves, as
well as give associated Weierstrass models, in the case of RM by the ring of
integers of $\mathbb{Q}(\sqrt{5})$. To prove this, we provide some techniques
for reducing quadratic forms over polynomial rings.
