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We consider continued fractions with partial quotients in the ring ofintegers of a quadratic number field $K$ and we prove a generalization to suchcontinued fractions of the classical theorem of Lagrange. A particular exampleof these continued fractions is the $\beta$-continued fraction introduced byBernat. As a corollary of our theorem we show that for any quadratic Perronnumber $\beta$, the $\beta$-continued fraction expansion of elements in$\mathbb Q(\beta)$ is either finite of eventually periodic. The same holds for$\beta$ being a square root of an integer. We also show that for certain 4quadratic Perron numbers $\beta$, the $\beta$-continued fraction representsfinitely all elements of the quadratic field $\mathbb Q(\beta)$, thus answeringquestions of Rosen and Bernat. Based on the validity of a conjecture of Mercat,these are all quadratic Perron numbers with this feature.

Finiteness and periodicity of continued fractions over quadratic number fields