The Dancing Mountains were nominated for a Fred Award in the Best Performance By an Inanimate Object or a Group of Inanimate Objects category in episode 412 of The Muppet Show. When they attempted to come forward to accept their award, they shook The Muppet Theatre. Kermit wisely decided to mail the statue to them.
Their approach was signaled by the musical refrain from The Sound of Music, an appropriate choice since these hills are very much alive.
The mountains may or may not be related to a hill upon which Grover sat that came alive to music.