There was exactly one point in time that any Emperor might have heard of some random peasant known as Jesus of Nazareth - and that would be if the regional governor thought it necessary to send a message to Rome, asking what to do about this man who's charged with inciting rebellion in an especially rebellious province. There was no reason for Augustus to know about - and I cannot emphasize this enough - a random carpenter in a small, backwater province on the edge of the Roman Empire. Even if (and I cannot emphasize enough that there is no reason why he should, since from the Roman Empire's perspective, the only important thing about this man may have been that he was inciting sedition and taking on divi filius) he did send anything about him in his correspondence, we have a small handful of letters from Augustus out of the probably hundreds of thousands that he wrote.

So probably not, nope.