With all of the commonness between names like Quintus, Germanicus, and Julius, I found myself wondering earlier today whether or not the Romans had and used nicknames commonly. I thought to myself 'surely they, or at least some, must've,' however, I'm wondering if there're any recorded uses of them.
The typical Roman name consists of praenomen, nomen, and cognomen. The praenomen would be what we consider a given name. The nomen was your family name, designating which clan or gens you belonged to. The cognomen began as a nickname, but evolved to become a hereditary name that specified which branch of the gens you were from. By the time of Caesar, these three names would have been your typical naming system, with an additional slot after the cognomen for the agnomen/nickname. A great example of an agnomen nickname is Publius Cornelius Scipio Africanus. Publius, from the Scipio branch of the Cornelii, called Africanus because of his role in the Punic Wars.
A different example of a non-agnomen nickname is Caligula. I'm not certain what this sort of nickname would have been called by the Romans, but it's pretty close to what we consider a nickname. Caligula means "little boots"--his father was a general, and as a child, the soldiers nicknamed him after the boots worn by soldiers. Caligula also had a hereditary nickname, Germanicus, from his father and grandfather's military campaigns in Germania.
Hope you find some of this helpful!