Can an off duty Cobb county Ga police officer arrest someone in Cherokee County Ga if they have not broken the law?

POSSIBLY. Not enough is known to answer the question.
Arrest them for WHAT? Was he serving a warrant? What was the arrested person charged with? What were the circumstances? Was the Cobb County officer a POLICE officer or were they actually a Deputy SHERIFF? It does make a difference as the two can have different jurisdictional authority.
Also, an arrested person will ALWAYS to claim they were arrested "for nothing."