It is true that, today, in the twenty-first century, Judaism is usually thought of as an ethnic religion and Jewish people are not generally known for their proselytizing. At least in the second and first centuries BCE and the first century CE, though, Jewish people in the Mediterranean world were far from totally disinterested in trying to convert other people to their religious practices and way of life. In fact, among ancient Greek and Roman authors in this period, one of the main things Jewish people became known for was their supposed habit of aggressively proselytizing.

Early Christianity's strong emphasis on proselytism is best understood not as a completely sudden new development or an example of early Christians doing something that no Jewish people had ever done before, but rather an example of Christians taking something that some Jewish people had already been doing and making it a major focus for their movement.

Jewish proselytism in antiquity

The epitomist Ianuarius Nepotianus in his epitome 3.2 summarizes a statement by the earlier Roman historian Valerius Maximus (fl. first century CE), who recorded in his De Factis Dictisque Memorabilibus 1.3.3 that, in 139 BCE, the Roman praetor peregrinus Cornelius Hispalus ordered for all Jews to be expelled from the city of Rome because the Jews were supposedly “Romanis tradere sacra sua conati,” which means “trying to transmit their own sacred rites to the Romans.”

If this report is accurate, it indicates that at least one Roman official at the time believed that the Jewish people living in the city of Rome were trying to convince Romans to follow at least some of their religious practices, if not outright convert them.

On somewhat more certain ground, we know that, in the late second century BCE, John Hyrcanus, the ruler of the Hasmonean Kingdom, conquered the region of Idumea located to the south of Judaea. He forced all the Idumeans to circumcise themselves and follow the Jewish law, mandating that anyone who would not follow the Jewish law would be forced to leave. The Jewish historian Titus Flavius Josephus (lived c. 37 – c. 100 CE) describes how this happened in his Antiquities of the Jews 13.9.1, as translated by William Whiston:

"Hyrcanus took also Dora and Marissa, cities of Idumea, and subdued all the Idumeans; and permitted them to stay in that country, if they would circumcise their genitals, and make use of the laws of the Jews; and they were so desirous of living in the country of their forefathers, that they submitted to the use of circumcision, and of the rest of the Jewish ways of living; at which time therefore this befell them, that they were hereafter no other than Jews."

This is one of the earliest known instances in human history of one nation forcing another whole nation to convert to their religion.

The Roman poet Quintus Horatius Flaccus (lived 65 – 8 BCE), commonly known today in English as "Horace," makes a famous joke in his Satires 1.4.142–43 in which he clearly plays on a popular stereotype that Jewish people will supposedly stop at nothing to convert people to their religion and rituals. He says:

"cui si concedere nolis,
multa poetarum veniat manus, auxilio quae
sit mihi (nam multo plures sumus), ac veluti te
Iudaei cogemus in hanc concedere turbam."

This means, in my own translation:

"If, for this, you are not willing to concede,
a mass of poets will arrive at my hand, they will be at my aid
(for we are by far the majority), and, just like
the Jews, we will compel you to concede into our throng."

A number of sources, including Josephus in his Antiquities of the Jews 18.81–84, the Roman biographer Gaius Suetonius Tranquillus (lived c. 69 – after c. 122 CE) in his Life of Tiberius 36, the Roman historian Publius Cornelius Tacitus (lived c. 56 – c. 120 CE) in his Annals 2.85, and the Greek historian Kassios Dion (lived c. 155 – c. 235 CE) in his Roman History 57.18.5, attest that, in 19 CE, the emperor Tiberius expelled many Jews from the city of Rome. Kassios Dion specifically claims that the reason for this expulsion was because the Jews were successfully converting so many Romans to their religious customs. He writes:

"τῶν τε Ἰουδαίων πολλῶν ἐς τὴν Ῥώμην συνελθόντων καὶ συχνοὺς τῶν ἐπιχωρίων ἐς τὰ σφέτερα ἔθη μεθιστάντων, τοὺς πλείονας ἐξήλασεν."

This means, in my own translation:

"When many Jews were congregating toward Rome and were converting multitudes of the locals to their own customs, he [i.e., Tiberius] expelled most of them."

The Roman Stoic philosopher Seneca the Younger (lived c. 4 BCE – 65 CE) rails at the apparent success of Jewish proselytism in Rome in a fragment from his now-lost treatise De Superstitione (i.e., On Superstition), which has been preserved through quotation by the Christian church father Augustinus of Hippo (lived 354 – 430 CE) in his apologetic treatise On the City of God 6.11. The fragment declares: “victi victoribus leges dederunt!” (“The conquered have given laws to the victors!”)

The Gospel of Matthew (henceforth abbreviated gMatthew), which was most likely written at some point between c. 75 and c. 95 CE, portrays the Pharisees, who were one prominent school of thought among religious Jews in the Levant in the first century CE, as particularly dedicated to the task of winning converts to their school of thought. gMatthew 23:15 portrays Jesus as denouncing them by saying (as translated in the NRSV):

"Woe to you, scribes and Pharisees, hypocrites! For you cross sea and land to make a single convert, and you make the new convert twice as much a child of hell as yourselves."

For more information about Jewish proselytism in antiquity, my main secondary source for this section is Louis H. Feldman's chapter "The Enigma of Horace's Thirteenth Sabbath" (pages 351–376) in his book Studies in Hellenistic Judaism, published by E. J. Brill in 1996.

(This answer is continued in the comment below.)