False. Jesus is mentioned twice by the historian Josephus.

Specifically, he says:

[...] so he assembled the sanhedrin of judges, and brought before them the brother of Jesus, who was called Christ, whose name was James, and some others [...]

(Antiquities of the Jews Book 20, Chapter 9, 1)

Before that he wrote:

About this time there lived Jesus, a wise man, if indeed one ought to call him a man. For he was one who performed surprising deeds and was a teacher of such people as accept the truth gladly. He won over many Jews and many of the Greeks. He was the Christ. And when, upon the accusation of the principal men among us, Pilate had condemned him to a cross, those who had first come to love him did not cease. He appeared to them spending a third day restored to life, for the prophets of God had foretold these things and a thousand other marvels about him. And the tribe of the Christians, so called after him, has still to this day not disappeared.

(Antiquities of the Jews, Book 18, Chapter 3, 3)

The second quote is a bit controversial, because historians agree it has been altered, making Josephus' judgement of Jesus more positive than what he had originally written. However, they also agree that the passage definitely mentioned Jesus and that he was executed by Pilate.

Furthermore, Christianity itself is often mentioned in Roman writings. Suetonius tells us that under Nero's reign:

Punishment was inflicted on the Christians, a class of men given to a new and mischievous superstition. He put an end to the diversions of the chariot drivers, who from immunity of long standing claimed the right of ranging at large and amusing themselves by cheating and robbing the people. The pantomimic actors and their partisans were banished from the city.

(The Lives of the Twelve Caesars, 16)

And last but not least, we know the Emperor Domitian declared Christianity religio illicita (impermissible religion) during his rule, sometime in the 80s AD. And we also have Pliny the Younger's letter dating from 112 AD to Trajan, asking the Emperor for counsel on how to deal with Christians in the province of Bithynia et Pontus, of which Pliny was governor.

In addition to the mentions by the historian Josephus that were already brought up, Christ and Christianity were also mentioned by the historian Tacitus.

Consequently, to get rid of the report, Nero fastened the guilt and inflicted the most exquisite tortures on a class hated for their abominations, called Christians by the populace. Christus, from whom the name had its origin, suffered the extreme penalty during the reign of Tiberius at the hands of one of our procurators, Pontius Pilatus, and a most mischievous superstition, thus checked for the moment, again broke out not only in Judaea, the first source of the evil, but even in Rome, where all things hideous and shameful from every part of the world find their centre and become popular.

Annals, Book 15

I just want to clarify your question, /u/MDDIY. It's more likely that the quote you recall didn't mention Christianity, only Jesus. It's a common trope among certain circles to argue against the existence of Jesus on the basis of a lack of evidence from outsider writers (i.e., non-New Testament). These same groups don't argue that Christianity didn't exist, because there is clear evidence that it did. (The top response in the present thread misunderstands your question slightly. Josephus was not a "Roman" in the sense meant by your quote, which means that there are indeed no "Roman" writings that mention Jesus.[1])

I would recommend heading over to /r/AcademicBiblical as well, which is jam-packed with scholars of the exact period involved in your question, and they can provide further detail if you're interested.

[1] With the debatable line in Tacitus a possible occurrence.

In all the known Roman writings, there is no mention of Christianity

This is quite an audacious statement considering the many Roman emperors who were Christians!

In addition to the mentions by Josephus and Tacitus of Jesus, the emperor Marcus Aurelius mentions Christians briefly:

What a soul that is which is ready, if at any moment it must be separated from the body, and ready either to be extinguished or dispersed or continue to exist; but so that this readiness comes from a man's own judgement, not from mere obstinacy, as with the Christians, but considerately and with dignity and in a way to persuade another, without tragic show.

Meditations Book Eleven

I don't know about Roman writings but Josepheus, a Jewish historian mentioned Jesus.