Let $K$ be a field and let $Q$ be a quaternion algebra over $K$. Then it is well-known that the class $[Q]$ of $Q$ in $Br(K)$ has order $2$. One can show this by constructing an explicit isomorphism $Q \otimes_K Q \cong M_2(K)$. My question is about the converse.

Does there exist a field $K$ and a division algebra $D$ over $K$ such that the class $[D]$ of $D$ in $Br(K)$ has order $2$, but such that $D$ is not isomorphic to a quaternion algebra over $K$?

If such a $K$ and $D$ exist, then it would also be nice to see an explicit example.

As a non-example, I believe that it follows from local and global class field theory that if $K$ is a local or global field, then every element of order $2$ in $Br(K)$ may indeed be represented by a quaternion algebra.