Does Autozone have trademark rights to TireZone?

AutoZone does not have automatic trademark rights to "TireZone" simply by virtue of its name. Trademark rights are generally established through usage in commerce and registration with the relevant authorities. If AutoZone has used "TireZone" in connection with its products or services and has registered it, then it could claim trademark rights. Otherwise, the rights would depend on whether "TireZone" is being used by others and if it is registered or being used in a way that could cause consumer confusion.