What can you do if your neighbor's dog damaged your property but he will not give you his insurance company's name?

If the dog did actual damage to your home, then you can contact your own home insurer and see if you can make claim that way. If the dog damaged a fence or a garden, your only recourse is to call the authorities or an attorney if the neighbors will not cooperate.

AnswerIf you don't want to place a claim on your own insurance (and besides, you will have to pay for your deductible), try suing him in small claims court and then he can turn it over to his insurance company.