The Temple of Fal'adora seems to still be a place that the Nightborne associate fondly with Elune. In the quest
The Valewalker's Burden, Oculeth says:
This, at least, means that they acknowledge Elune's existence and some semblance of power.
However, Highborne (and former Highborne) in general still seem to believe in Elune, but they don't outright worship her. High Elves named their city after her: Silvermoon. Dath'remar Sunstrider did seem to have some respect for the goddess in the War of the Ancients - he saved Tyrande because she was the High Priestess, and would later found and name Silvermoon. There are other moon motifs in Quel'thalas as well. But, we've seen no evidence of any High/Blood Elf that actively worships Elune. Likewise, the Highborne of the Shen'dralar haven't had any mention of actively worshiping Elune. The closest we've seen is in the pre-Cataclysm quests introducing the Highborne and Sunwalkers, the leader of the Highborne was looking "longingly" at a moonwell - but more for the power within it.
Nightborne, meanwhile, don't have any temples in Suramar city itself. The closest were Fal'adora and the Tomb of Sargeras - which was originally connected to the Nighthold 10,000 years ago. But, even still, you'd think there would be smaller places of worship inside the city. There are moon motifs, just like in Quel'thalas, but Oculeth is the only one to speak about Elune (perhaps Thalyssra mentioned it once in regards to Tyrande?).
As a nation, no, it seems they are like the ancient Highborne, who may believe Elune exists, but they don't outright worship her themselves. Nightborne priests could be a smaller subsection that still openly worships Elune, however. There isn't really any reason why they can't or shouldn't - even with the shield up, they had an observatory to view other planets and celestial bodies. Or, they could go with the more "shadow" aspect of the Nightwell's former power, and (in lore) exclusively have shadow priests.