  In the late '60s, B. Dwork studied a Frobenius structure compatible with the
classical hypergeometric differential equation with parameters
$\left(\frac{1}{2},\frac{1}{2} ; 1 \right)$ by analyzing behavior of solutions
of the differential equation under Frobenius transformation. Recently, P.
Scholze conjectured the existence of $q$-de Rham cohomology groups for any
$\mathbb{Z}$-scheme. In this paper, we give a Frobenius structure compatible
with the $q$-hypergeometric differential equation with parameters
$(q^{\frac12},q^{\frac12};q)$ by showing a $q$-analogue of some results of
Dwork. This construction gives a $q$-deformation with
$(\varphi,\Gamma)$-structure over $\mathbb{Z}_p[[q-1]][[\lambda]]$ of the de
Rham cohomology of the $p$-adic Legendre family of elliptic curves which has
Frobenius structure and connection.
