  The properties of continued fractions whose partial quotients belong to a
quadratic number field K are distinct from those of classical continued
fractions. Unlike classical continued fractions, it is currently impossible to
identify elements with periodic continued fraction expansions, akin to
Lagrange's theorem. In this paper, we fix a real quadratic field K and take an
ultimately periodic continued fraction with partial quotients in
$\mathcal{O}_K$. We analyze its convergence and the growth of absolute values
of $\mathcal{Q}$-pairs, and establish necessary and sufficient conditions for a
real quartic irrational to have an ultimately periodic continued fraction
converging to it with partial quotients in $\mathcal{O}_K$. Additionally, we
analyze a specific example with $K=\mathbb{Q}(\sqrt{5})$. By the obtained
results, we give a continued fraction expansion algorithm for those real
quartic irrationals $\xi$ belong to a quadratic extension of $K$, whose
algebraic conjugates are all real. We prove that the expansion obtained from
the algorithm is ultimately periodic and convergent to the specified $\xi$.
