So after looking deeper, both are Atayal.
The one that uses the tuqi spelling for road is likely from the Kinhakul (Squliq) branch. Languages from this branch refers to people as Squliq and road as tuqi.
The one that uses the tu’iy spelling for road is likely from the Mthu’ (S’uli’) branch. 4 languages belong to this branch and they refer to people as s’uli’ and road as tu’iy.
There are two more Atayal sub language branches, the Mkmuyaw and the Plngawan. The later two are mostly used by people living on the West side of the central mountain range. So it makes sense that in Yilan only the first two sub-branches are written on road signs.