At that time, (first century CE) there really were Latin speaking Romans in Jerusalem and administrating the region they called Judea. This is before the larger cultural and then eventually political split of the empire, so your average centurion would likely still be speaking Latin in order to communicate with his superiors and other troops.
Of course, they could definitely have spoken other local languages as well (depending on their personal history and length of stay).
Edit: Edited to add the date and also wanted to say that while your average centurion and likely a lot of the soldiery would be able to speak (or at least understand) Latin, the aristocracy and administration of the province would likely be able to speak Greek as well, or at the very least communicate through proxies in Greek and Hebrew.
Latin is really interesting in its history. While it was a very large part of Roman culture and certainly still influences other languages today, it was mostly an Italian thing during the Empire. Romans from the Italian peninsula and, later on, the provinces closer to Italy, were the primary speakers of Latin in day to day use. Outside of that, the language was quite widespread, but was more of an administrative language, as others have said.
With imperial Rome’s penchant for integrating/melding itself with conquered cultures, and the influence of Latin through the Roman Catholic Church in the middle ages, Romance languages of Europe can really be seen more as expressions of Latin through different existing linguistic traditions.
I have heard that in some ways, Spanish is one of the closer varieties to Imperial Latin as, between the Roman conquering of the Iberian tribes and the eventual invasion of the Visigoths during the fall of the Western Roman Empire, it was one of the most culturally and administratively stable regions of the empire with much of its population being quite proud to consider themselves a part of the empire. Two of the more famous emperors, Trajan and Hadrian were both from Hispania and the region was seen as somewhat of a model province. When the Germanic Visigoths took over, they were already dealing with a pretty strong linguistic tradition in the region.
French is sort of another Germanic take on Latin. The Franks really started picking up Latin in the middle ages. Like the majority of Europe, the Franks brought Latin influences in primarily through Christianization and Roman Catholicism’s power structure, prestige, and connection to the somewhat mythical days of the empire.
Both of these languages evolved from continual and persistent contact with a conquering empire and religious structure where Latin was seen as a connection to power and importance. Latin’s place in other European languages comes largely from those connections as well and I would say, the strongest connections come from the religious framework of the Catholic Church.
The Eastern Roman/Byzantine Empire, by contrast, had already been leaning more heavily into Greek language and culture, even from the earlier days of the empire. This is mostly due to geography and preexisting cultures, but also due to the Eastern Empire’s ability to operate more like it had before the split. As they controlled the wealthier and more productive provinces, Byzantine rulers initially had more resources at their disposal and less of a need to create/capitalize on the mythological prestige of an old imperial language and religious structure to bring together disparate peoples.
Instead of promoting a more cohesive cultural/religious framework with outside civilizations, rulers in the East relied more on wealth and payment to pit rivals against each other or just bribe warlords to go away for a while. This would eventually lead to the slow retreat of Eastern borders and the conquest of important provinces by the more unified Arabs and Turks as allies became harder to find and the money ran out.
Basically, by the time Latin resurfaced as a prestigious language in Western Europe, it had already been largely abandoned in the surviving Eastern Roman Empire. With less linguistic tradition of Latin in the region, the cultures that took over after the fall of the Eastern Roman Empire did not need to engage with the language much past scholarly curiosity, and even then, most of the important works were already written in Greek to begin with.
I would argue that due to the nature of the Roman Empire, Latin struggled to spread (especially Eastward and Northward) even during its heyday and was largely resurrected as an administrative language in Western Europe due to Germanic warlords wanting to have a connection to past glories of the Roman Empire and the potentially lucrative economic system of the Roman Church. It really is amazing how far it has reached linguistically since then. We still collectively treat it with reverence in academic and legal contexts millennia after it was spoken in day to day usage.