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

On Periodicity of Continued fractions with Partial Quotients in Quadratic Number Fields