The extendibility of the parametric $D(-1)$-triples

Let $N$ be a positive integer such that $4N^2+1=q$, where $q$ is a prime. In this paper, we prove that the Diophantine $D(-1)$-triple of the form $\{1,4N^2+1,1-N\}$ cannot be extended to a quadruple in the ring $\mathbb{Z}[\sqrt{-N}]$, with a non-square integer $N＞2$. If $N＞2$ is a square, then $4N^2+1$ is not a prime, and the set $\{1,4N^2+1,1-N,1+N\}$ is a $D(-1)$-quadruple in the ring $\mathbb{Z}[\sqrt{-N}]$, thus also in the ring of the Gaussian integers as well.