I don't understand your reference to the model (since the geometry of the hyperbolic plane does not depend on any model), but, in fact, the Beltrami-Klein model demonstrates that any qualitative statement about convex sets in the Euclidean plane holds in the Hyperbolic plane and *vice versa*, since the model maps convex sets to convex sets.

**EDIT** This has (almost) absolutely nothing to do with the above, but the topological version of Helly's theorem goes back to *at least* Debrunner (and it is a Monthly paper, so is human-readable), no need to allude to Farb's paper.

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/wSpBf.png