I think to answer this question, we have to work backwards. Officially, diplomatic relations between Serbia and Israel began in 1992. That’s a bit late, but at the time they were part of the Federal Republic Yugoslavia. However, they were the fifth country in Europe to recognize Israel. It’s a bit more complicated though as the government of the Kingdom of Serbia was in exile at the time when they made the official recognition. The political relations between the two have been a bit controversial as well, as in 1995, Israeli weapons were found among Serbian militants in Bosnia. And it was reported that Israel had purposely supplied them weapons because of their pro-Serbian bias.
But we can move further back. A good and easily accessible source for this is the Jewish Virtual Library. To be fair, the source is a bit biased as a whole, but the history they provide is pretty straight forward. The earliest point they place Jews in Serbia is the 12th century. At that time, Jews, up until the 18th century, were treated quite well. The main profession at that time for Jews was selling salt, and they were largely responsible for very large trade networks.
Starting in 1804, the Serbs began a war with the Turks in order to gain their own independence. The Jews sided with the Serbs and even supplied them with weapons. When the Serbs finally won in 1830, the friendly relations were largely gone. After that, Jews were regulated truly to a second class and were prohibited from many professions. In 1856 the Jews were expelled from provincial towns. And Jews during the Holocaust were persecuted there as well. So it was ups and downs.
So can we get back before the 12th century? The World Jewish Congress states that there are traces of Jewish presence in Serbia back to Roman times. They don’t specify what they mean by Roman times though. This is what the European Jewish Congress also states, and it is what you will find on Wikipedia. However, the European Jewish Congress does state that there is no evidence of continuity until the 15th century. Most sources agree that there is little evidence until the 10th century.
I will get back to the history of Jews in Serbia in just a bit, but I think an aside is useful here. In the town of Zemun, in Serbia, the precursor of the modern Zionist movement began by a Sephardi rabbi named Judah Ben Shlomo Hai Alkalai. In my opinion, Alkalai is an understudied figure. But we do know a bit. He came from a prominent family of rabbis. He spent most of his adolescent years being educated, and at 27 years old, he became the communal rabbi of Zemun. He was the Rabbi for both the Sephardi and Ashkenazi Jews in the small town. In 1834 he produces his first work called “Hear O Israel.” He reinterprets the well-known Jewish prayer by the same name (the Shema Yisrael). He sees it as a commandment that all Jews should be together as one. He advocated for a national congress of sorts that would see the return of the Jewish people to the historical Israel. After things get worse for Jews in Israel, he begins pushing this even more. But his focus was mainly the unity of the Jews.
Theodor Herzl, who is seen as the visionary behind the modern Zionist movement also has ties back to Zemun and Alkalai. Herzl grandfather and father were both active in the Jewish community in Zemun, were tied to Alkalai, and really developed the seeds of Zionism in Herzl. So one could make the argument, and some have, that Zionism and really the modern Israeli state were created in Serbia. Which could explain the deep tie there.
Back to the Jews in Serbia. A great work on the history of the Jewish diaspora is the book, Jewish People, Jewish Thought. During the Roman times, Jews would be dispersed quite a bit. After the second Jewish War, Rome really cracked down on Judaism, and that forced many to leave Palestine. By the later Roman period, you find important Jewish centers in places such as Italy, and even some large settlements in France. There are smaller towns in modern-day France (Gaul) and near Roman garrison towns along the Rhine. At the same time, there were Jewish communities throughout Greece.
Modern day Serbia was also under Roman control, at least in part, starting around 168 BCE. This conquest would expand to include pretty much all of Serbia proper (basically the core of modern Serbia). After the western Roman Empire fell, the Byzantines (the eastern Roman Empire) controlled portions of the area until 610 C.E. So exactly what is meant that by Serbians and Israelis having connections from the Roman period isn’t exact. And it wasn’t known as Serbia at that time. It wasn’t until the 6th century that the Serbians began holding part of the area. So we are looking at more of the Byzantine period.
So, by the end of the late Roman period (and here we are talking about the western empire), Jews were in communities around what is modern day Serbia. But evidence that they moved into the Serbian area is scarce. It’s possible that there were small communities in the area, even before the Serbs arrived, but that is more speculation as we lack hard evidence. And most sources, if they make this claim (I don’t know of any purely scholarly sources that make the claim) also state that there is little evidence for settlement up until the 10th century.
If I had to guess, I would say that the claim is more one of exaggeration based on a real importance with the Zionist movement having a foundation in Serbia. And it’s also a stretch to say it was a friendship because officially, the Serbian government persecuted Jews once they gained their independence.