It is well known that there is a bijection between $GL_2(\mathbb Z)$-equivalence classes of binary quadratic forms with $\mathbb Z$ coefficients of a fixed discriminant $D$, and the narrow class group of $\mathbb Q(\sqrt{D})$.
Suppose $K$ is a number field, does there exist a similar correspondence between $GL_2(\mathcal O_K)$-equivalence classes of quadratic forms with $\mathcal O_K$ coefficients of some discriminant $D$, and the class group of the quadratic extension of $K$ with the same relative discriminant?
I suspect this may not be true in general, since relative discriminants would be an ideal and not an element of $\mathcal O_K$ in general so the correspondence would be a bit murky. But perhaps if extra conditions were imposed on $K$, e.g. $K$ has a trivial class group so $\mathcal O_K$ is a PID, then a similar correspondence might exist?