It is extremely unlikely that you would be on the hook for this. And it's even less likely that your insurance will cover it, since the injury took place on someone else's property and was not caused by the condition of your property. Your neighbor's insurance should cover it, or the neighbor can sue the former tenant. My answer would be different if the damage was caused by a problem with your property itself--a rotting tree that fell on the neighbor's house, or water damage caused by a broken pipe on your property--but damage caused by a tenant should not result in liability for you. But, you could always call your insurance agent and ask. Who knows? You might have some weird kind of policy that includes this sort of thing, and if so, it will help "mend fences" with your neighbor.
NOTE: This answer is intended for information and educational purposes only and is NOT legal advice. Advice on specific legal matters requires an in-depth discussion of facts that can only be done in private, which is not possible through a website accessible to the public. Also, note that I am only licensed to practice law in Michigan, am not familiar with the laws of most other jurisdictions, and could not offer legal advice (even if I wanted to do so) outside of Michigan.