The name 'Moria' was first used in around III 1980
The Balrog and Orcs that infested it were largely destroyed in the War of the Ring (III 3018 to III 3019), after which the name would likely have fallen out of use.

The central regions of the Misty Mountains, to the northwest of Lórien

Orcs, Trolls and one Balrog (Durin's Bane)

mo'ria

'Black (or Dark) Chasm'

The Black Chasm, The Black Pit

The name 'Moria' is often used as a synonym for Khazad-dûm. This is not strictly correct - 'Moria' was not used of the ancient Dwarf-city until after it was deserted by the Dwarves themselves - but by the late Third Age the two names had come to be used interchangeably.