Were they close trading partners? Enemies? Also, what were the demographics of the Syria Palaestina province of the Roman Empire?
Some Basic History
There were for a while, many Jews living in Arabia. Historians estimate that up to 80,000 Jews had moved into Arabia after the destruction of the First Temple. These Jews mainly lived at the oasis of Khaybar, Mecca, and Medina.
After the battle of Badr, between the Muslims of Medina and the Quraish of Mecca, many Jewish clans broke treaties and deals with Mohammed. Mohammed then used this oppurtunity to attack and expel some of the largest Arabian Jewish clans in Arabia.
Caliph Umar, Mohammed's successor, decreed that Jews and Christians should be expelled from all but the southern parts of Arabia. Due to this, many Yemenite Jews still existed through the Ottoman Era.
Starting around 500 AD, a group of Jewish merchants, known as the Radhanites, were very prevalent in trade between Europe and the Islamic World. Due to this trade, Jewish merchants became respected in the Islamic World, from before the conquest of the Levant through the Middle Ages.
When the Rashidun Caliphate conquered the levant, Jews and Christians were tolerated, and some even had better lives than when they were living under Byzantine rule.
So, Jews and Arabs have had a very long history:
Prior to the birth of Mohammed, Jews and Arabs were different ethnic groups that traded with each other and lived very close to each other.
After the birth of Islam, Jews were treated differently in different circumstances, including being tolerated by the Rashidun Caliphate, and being expelled by Mohammed and his followers.
Addressing Demographics
Syria Palaestina was a large province. In the northern portion, the population consisted of mainly Phoenicians, Greeks and Romans, while their were some minorities of Ismaelite Arabs, such as the Itureans. In the southern portion, until around 200 AD, Jews were the majority of the population. As the Jewish community declined, Samaritans and Greco-Romans became the dominant societies in the area.
At the time of the Byzantine period, Jews were a minority, and they were living alongside Samaritans, pagan Greco-Syrians and a large Christian community.
Some scholars say that Islam did not become the majority religion of Palestine until at least the 9th century and possibly even as late as the Mamluk era (1250–1516). The majority of the Muslims arrived with and after the Muslim conquest, and Islam spread very quickly in Arabs, Christians, and some Jews and pagans.