The relationship, if you want to call it so, would be rather far-fetched. In TTS you normally investigate the tunneling of electrons, not protons. Now, in proton transfer processes the tunneling effect is a relevant subprocess, so if you model proton transfer reactions by theoretical methods, a tunneling correction to rate coefficients would be required or your calculated rates would be too slow. I haven't done such calculations myself since 2012, but even back then all the softwares I used had this option available.

The physics behind proton and electron tunneling is in both cases the quantum mechanical tunneling effect, but to my extent of knowledge that's as far as the relationship goes.