No, and there are a couple of reasons for this.
The first has to do with where the insurance responsibility falls with downed trees. Generally you file a claim with your own home insurance company if a tree falls and damages your property — no matter where the tree came from. So if the tree caused damage in your neighbor’s yard, your neighbor would file a claim with his own insurance company.
The neighbor’s insurer could come after you and your home insurance company, though, if there were signs you neglected caring for the tree, and this led to its fall.
Secondly, home insurance typically does not cover the removal of fallen trees if they don’t damage property. To qualify for a claim, the tree must have damaged a structure covered by the policy. An exception is sometimes made if a tree blocks a driveway or a wheelchair ramp for accessing the house.
What if the tree had caused damage to your neighbor’s house? In that case, your neighbor’s home insurance policy would pay for the damage, minus the deductible, and the cost of removing the tree up to a specified limit.
Although home insurance probably won’t come into play here, that doesn’t you can’t be neighborly and offer to pitch in to remove the tree.