While fences are covered by homeowners insurance, the damage must be caused by a named peril. In your case, the repairs will likely not be covered.
Fences are covered by the other structures coverage of your homeowners insurance policy, which also covers things like pools and sheds. However, the damage must be due to a named peril. Named perils are all sudden, unexpected causes of damage, like a fire, storm, or theft.
Wear and tear, age, or poor maintenance are never covered. You’ll need to pay for the fence repairs yourself.