I want to give an update on the situation. The insurance company investigated the situation and paid the neighbor $175. The insurance company called to tell me that the neighbor had called them very upset and they were letting me know that he told them that he was going to sue me.
However, there's been more that's happened. My dog has had the opportunity to attack other birds, mostly pheasants, in my yard and the dog will chase them but she never catches them even when she can. She swerves away at the last second. The neighbor confided to a common friend that the bird enclosure had been left open accidently all night when his birds had been killed and we live in an area with lots of coyotes, dogs and other animals, even wolves. Come to find out my neighbor didn't actually see my dog kill their birds, he saw the dog in the bird enclosure with the dead birds and he just assumed the dog killed the birds. There's no doubt my dog was in the enclosure, but there's a very real possibility the birds were dead before my dog arrived. Another neighbor has had birds killed, too, and it certainly wasn't my dog that did that.
At this point I am waiting to see if I get sued.