Don't know about Mexico, but this is what I found on the belize.net forum - just do a search for dog.
4/12/2001 She will need to get a permit to import her pet from a vet in Belize. Here's information about a vet that I contacted when we took our dog with us to Belize last fall. I spoke with him by email and on the phone. If he doesn't reply to email within a week, I'd phone him. He's really nice and helpful (so is his wife Miriam, who I also spoke with on the phone). Michael DeShield,Animal Medical Centre, phone 501-2-33781 or [email protected]
She will also need to get a Health Certificate from her current vet which lists all the vaccinations and the current condition of the animal. I believe these need to be dated no more than 72 hours of the date when she begins her journey with the dog (we drove from Madison, WI so the trip took 7 days). I also brought all my dog's medical records with me, just in case I needed to get any treatment for her while we were in Belize. At customs at the border we were never asked for any information, even though I kept asking if they wanted to see it, it just seemed that as long as I had the stuff it was fine. I wouldn't risk it though and just bring whatever Dr. Deshield thinks she needs. Good luck!