  Given a quartic Galois extension $L/\mathbb{Q}$ of number fields and a
Hopf-Galois structure $H$ on $L/\mathbb{Q}$, we study the freeness of the ring
of integers $\mathcal{O}_L$ as module over the associated order
$\mathfrak{A}_H$ in $H$. For the classical Galois structure $H_c$, we know by
Leopoldt's theorem that $\mathcal{O}_L$ is $\mathfrak{A}_{H_c}$-free. If
$L/\mathbb{Q}$ is cyclic, it admits a unique non-classical Hopf-Galois
structure, whereas if it is biquadratic, it admits three such Hopf-Galois
structures. In both cases, we obtain that freeness depends on the solvability
in $\mathbb{Z}$ of certain generalized Pell equations. We shall translate some
results on Pell equations into results on the $\mathfrak{A}_H$-freeness of
$\mathcal{O}_L$.
