I found the below link to calculate hyperbolic distance in hyperboloid model but I am looking is this formula is invariant even after performing lorentz transformation on the coordinates
What's the right way to calculate hyperbolic distance on the hyperboloid model?
Yes, Lorentz transformations prserve distances, they describe the isometries of the hyperbolic plane.
As discussed in the question you referenced, the distance between two points $u$ and $v$ depends on the bilinear form $B(u,v)$ which can be written as
$$B(u,v)=u^T\,\eta\,v \qquad\text{with}\quad \eta=\begin{pmatrix}1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}$$
Now a Lorentz transformation is defined as a transformation which does not alter the bilinear form. A Matrix $\Lambda$ is a Lorentz transformation if $\Lambda^T\,\eta\,\Lambda=\eta$. Which means $B(\Lambda u,\Lambda v)=B(u,v)$ and therefore the distance of the transformed points has to equal the distance of the original points.