I think $K$ has to be an abelian extension of $\mathbb{Q}$ of class number one. Because if the Hilbert class field is abelian over $\mathbb{Q}$, then it is contained in a cyclotomic field, so $K$ itself is inside that cyclotomic field so $K$ is abelian, but an extension of fields inside a cyclotomic field is always ramified (not 100% sure here but I think it's OK) so the Hilbert class field cannot be a proper extension of $K$.

I think $K$ has to be an abelian extension of $\mathbb{Q}$ of class number one. Because if the Hilbert class field is abelian over $\mathbb{Q}$, then it is contained in a cyclotomic field, so $K$ itself is inside that cyclotomic field so $K$ is abelian, but an extension of fields inside a cyclotomic field is always ramified (not 100% sure here but I think it's OK) so the Hilbert class field cannot be a proper extension of $K$.

Edit: Definitely not OK. See comments below.