The very first instance that I can find, in which camels were explicitly used by Roman allies was in Caesar's Alexandrine War, where a local king in northwest Arabia (Malichus I.) sent a troop of camel riders (later called "Dromedarii" in Latin) as an auxiliary force to aid Caesar. Caesar calls them simply "equites" (riders) making no special mention of them being camel riders, but given their origin they most certainly were. As the Empire expanded into the east, camels became better known in the Roman world, mostly used as beasts of burden, even though Roman allies and tributaries in North Africa and the Levantine also used them militarily. Finally under emperor Trajan (reigned 98 - 117) we have the first record of a Roman cavalry unit made up of camel riders, now officially called "Dromedarii". They were one "ala" strong, meaning something between 500 and 1000 riders, and were probably used in Roman provinces bordering dessert.

An important thing to remember is however, that they were not a regular part of the Roman army. Judging from some written accounts about their equipment and recruitment, it seems that there was probably never a single "Roman" (as in, a person being born in Rome or even Latium) who was trained to ride and fight on a camel. The Romans used the military experience of local populations, who, in Syria and Arabia, had been using camel cavalry for centuries and knew how to fight on them effectively. Dromedarii were mostly used on patrol or guard duty where horses were rare or ill-suited for the terrain or climate. This is why - at least after a quick look through some sources - I can't tell you about any important battles in which Dromedarii played a major role. The writer Vegetius even seems to say they "inefficax bello est" (are inefficent in war), though I have to admit that I haven't seen this source myself and am just quoting it from "Dabrowa, Dromedarii in the Roman army: a note"

edit: Some spelling, some phrasing mistakes