  Groups of order $4$ are isomorphic to either $\mathbb{Z}/4\mathbb{Z}$ or
$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. We give certain
sufficient conditions permitting to specify the structure of class groups of
order $4$ in the family of real quadratic fields $\mathbb{Q}{(\sqrt{n^2+1})}$
as $n$ varies over positive integers. Further, we compute the values of
Dedekind zeta function attached to these quadratic fields at the point $-1$. As
a side result, we show that the size of the class group of this family could be
made as large as possible by increasing the size of the number of distinct odd
prime factors of $n$.
