Yes, damage to your fence is covered in the event of a storm with wind force 7 or higher according to the KNMI (Dutch meteorological institute). View  the terms and conditions of our home insurance .
If you share a fence with your neighbours, your insurer and your neighbour's insurer will pay 50% each. If you rent your home or you own a home that is part of an Homeowners' Association (VvE), storm damage to your fence is covered under the tenant's or owner's interest of your home contents insurance. If the neighbour's fence falls and causes damage to you, you can claim the damage through your neighbour's liability insurance policy.
If you are having problems with the neighbour's insurance company, you may be able to receive help through your legal assistance insurance in order to recover the damage from your neighbours.