Legal Question in Traffic Law in Arizona

I am sorry to say that the simple answer is "no". A police officer can check your license plate status for no reason or any reason. License plates are considered to be open for public viewing and therefore generally not subject to an expectation of privacy. In fact, there's an appellate case with similar facts as yours (officer ran the person's license plate for no reason, or on a hunch). United States v. Diaz-Castaneda, 494 F.3d 1146, 1148 (9th Cir. 2007) ("We are presented with a question this court has not yet resolved: Does a license plate check by a law enforcement officer that reveals information about a person's car ownership, driver status and criminal record constitute a search under the Fourth Amendment? We agree with all the other courts that have considered the issue that it does not."). Google the case name and you can learn more.