Only if Neighbor A was negligent. If A was not negligent, then B's insurance pays.
https://www.allstate.com/tools-and-resources/home-insurance/tree-falls-on-house.aspx
Q: What happens if the tree was on my neighbor's property?
A: If your home is damaged by your neighbor's tree, the III says you'll typically find that your homeowners insurance may help pay to repair the damage to your house (or other structure, if the tree falls on your fence, for example).
Q: Am I responsible if a tree from my property damages my neighbor's property?
A: You're typically only considered responsible if
neglect on your part was a contributing factor to the tree's demise. If not — say a storm knocked your healthy tree onto your neighbor's house — your neighbor will likely have to file a claim through his or her own insurance.