Greetings from Alaska but I lived in Billings with my 3 greyhounds. I don't know what part of Billings you are in but the vet I used was Lockwood Veterinary Service, 3025 Old Hardin Rd. (406) 252-7392.
It's a small clinic and they get to know you and your animals. It was a bit of of a drive for me, and there were vets closer but I feel it was worth the drive.
If you go there, tell them I sent you!
Let me know if you want to connect with other greyhounds & their owners.