I know that every isometry of the sphere $\Bbb S^2$ is the restriction of some $A \in {\rm O}(3,\Bbb R)$: namely, if $A_0:\Bbb S^2\to \Bbb S^2$ is an isometry, then $A_0 = A\big|_{\Bbb S^2}$ where $$Ax = \begin{cases} \|x\| A_0\left(\frac{x}{\|x\|}\right), \text{ if }x \neq 0 \\0 , \text{ if }x=0\end{cases}$$Then I thought of isometries $\Lambda_0:\Bbb H^2 \to \Bbb H^2$, where I see $\Bbb H^2$ inside Lorentz-Minkowski space $\Bbb L^3$. I'd guess that every such $\Lambda_0$ is a restriction of some orthochronous Lorentz transformation $\Lambda: \Bbb L^3 \to \Bbb L^3$, and I add the orthochronous condition so that $\Lambda$ doesn't swap the connected components of the two-sheeted hyperboloid.
Trying to define $\Lambda$ from $\Lambda_0$ mimicking the construction for $\Bbb S^2$ doesn't work immediately for lightlike vectors. Every lightlike vector can be approximated by timelike vectors, so I think that this approach is fixable, but I'd like to see that details addressed.
Where can I find the proof of that (if the result is actually true)? Thanks.
Yes, this is correct. To extend an isometry of $\mathbb{H}^2$ to one of $\mathbb{L}^3$, one approach is to note that the isometries of Minkowski space are in fact linear transformations of the underlying vector space, so that once you know how they act on three linearly independent vectors, you know how to extend to all of $\mathbb{L}^3$.
(You can also define the action on spacelike rays by orthogonality and the action on lightlike rays by a limiting procedure, but I think exploiting the linear structure is much cleaner.)
PS- This MO post explains why Lorentz transformations are necessarily linear.