A similar question was asked a while back. I'll provide the link below.

The short answer is not really. The Roman nobility were identified by there nomen and cognomen. For example Gaius Julius Caesar belong to the gens Julia and the smaller Caesar family group. They didn't have the type familial emblems like later heraldry, but on coins their heritage could be identified by the nomen and cognmen inscribed on them.

Furthermore, there were particular symbols different gens may have drawn upon. For example artwork associated with the Julia often draws upon the mythic founders of their line such as Ascanius, Aeneas and the goddess Venus (in particular the aspect Venus Genetrix).

/u/jsinnottdavies https://www.reddit.com/r/AskHistorians/comments/2ieg1r/did_the_romans_have_anything_like_coats_of_arms/

While Rome didn't have any kind of institutional familial branding, Caesar did use the elephant to represent either his family or himself. The coins were minted as troop payment during his Gallic campaign. Caesar used a folk etymology for the origin of his cognomen, that it derived from the Punic word for elephant, with the story that an ancestor had dispatched one of these beasts during the Punic wars.

Another possibility is that the image comes from his using the animal in his invasion of Britain. This comes from a centuries later compilation of military tactics, and I'd bet is more tall tale as Caesar attracts Hannibalistic comparisons. It seems too much an accomplishment in itself for Caesar not to mention it in his own account of the invasion, after all.