As in, members of the elemental plains, no, but nor are dragons. As well, just like the ones mentioned, the dragons are not identified in-game as elementals either.
Found the answer, and it is no. They are not elementals anymore, their predecessors were, ages ago, and their spawn evolved into flesh and blood over time.
The proto-dragons descend from elementals who avoided being imprisoned in the Elemental Plane and instead remained on Azeroth, where some of them gradually transformed into creatures of flesh and blood. More specifically, Brann Bronzebeard recently uncovered evidence, corroborated by reports from adventurers in Deepholm, that proto-dragons may have origins in elemental drakes such as stone and storm drakes. The inhabitants of Deepholm, the Skywall, the Firelands, and the Abyssal Maw are less than talkative on these matters, however, and most of them were not around when the elemental prisons were created.
The proto-dragons are thus early version of dragons before they evolved into the sapient, beautiful majestic creatures they are known as today.
(Source)