Short answer, no, we don't know. The exact route Hannibal took has been a point of contention for centuries. This map does a good job of showing the various possible routes.

Interestingly enough just a few months ago Archaemetry published some new research which uncovered horse gut bacteria consistent with the era on the Traversette pass (the southernmost option on the map). Basically it's in the right place, from the right time, and in sufficient quantity to account for the right number of horses. It isn't conclusive proof, but it is suggestive. Plus, Traversette was already one of the most popular options anyway. So at the moment Traversette has to be considered the leading candidate, but it's still far from settled.