  In this paper, we give an extension of the classical story of the elliptic
modular function to the Hilbert modular case for $\mathbb{Q}(\sqrt{5})$. We
construct the period mapping for a family $\mathcal{F}=\{S(X,Y)\}$ of $K3 $
surfaces with $2$ complex parameters $X$ and $Y$. The inverse correspondence of
the period mapping gives a system of generators of Hilbert modular functions
for $\mathbb{Q}(\sqrt{5})$.
  Moreover, we show an explicit expression of this inverse correspondence by
theta constants.
