Mount Everest was a mountain on Earth, part of the Himalayas.
In 2152, Trip Tucker recalled standing on an asteroid crater twice as high as Everest to Zho'Kaan while they were stranded on a moon together. (ENT: "Dawn")
Gideon Seyetik believed Da Vinci Falls on Blue Horizon to be the same height as Everest. (DS9: "Second Sight")
In the Julian Bashir, Secret Agent holoprogram, Everest was the location of Hippocrates Noah's hideaway. It was on the southeastern slope at an elevation of twenty five thousand feet. Noah had it built there, as he planned to flood the Earth and start the Human race anew. (DS9: "Our Man Bashir")
Mount Everest was referenced in the final draft script of ENT: "Terra Prime", as it stated, "
Conditions on partially terraformed Mars
are the equivalent of being half a mile higher than Mount Everest."