If the trail is inside a state park, firearms are prohibited in the state administrative code (I don't remember the exact reference atm). National Parks are off-limits too... at least until early next year.
Otherwise, you are fine to carry a handgun regardless of any local ordinances (see CoA Section 11-45-1)
IMO (IANAL) a bicycle isn't considered a vehicle for the purposes of 13A-11-73, so you should be fine to carry unconcealed without a permit. If you have a GFL than you don't have to worry about carrying in vehicles/concealed.