  Let $d\equiv 2\pmod 4$ be a square-free integer such that $x^2 - dy^2 =- 1$
and $x^2 - dy^2 = 6$ are solvable in integers. We prove the existence of
infinitely many quadruples in $\mathbb{Z}[\sqrt{d}]$ with the property $D(n)$
when $n \in \{(4m + 1) + 4k\sqrt{d}, (4m + 1) + (4k + 2)\sqrt{d}, (4m + 3) +
4k\sqrt{d}, (4m + 3) + (4k + 2)\sqrt{d}, (4m + 2) + (4k + 2)\sqrt{d}\}$ for $m,
k \in \mathbb{Z}$. As a consequence, we provide few counter examples to a
conjecture of Franu\v si\'c and Jadrijevi\' c (see Conjecture 1.1).
