The class number one problem for the real quadratic fields $\mathbb{Q}\left(\sqrt{(an{)}^2+4a}\right)$

We solve unconditionally the class number one problem for the $2$-parameter family of real quadratic fields $\mathbb{Q}(\sqrt{d})$ with square-free discriminant $d=(an{)}^2+4a$ for positive odd integers $a$ and $n$.