Blue Mountains: known as Ered Luin in Sindarin.
Mines of Moria – Moria is Sindarin for Black Abyss. Also known as Khazad-dûm in Khuzdul (meaning Dwarves’ Delvings/Mansions) and Dwarrowdelf (archaic Westron for Dwarf Delving)
The dwarves fled Moria after awakening the Balrog and settled in the Grey Mountains, where, due to the dragons gaining strength, they again had to flee and settled in the Lonely Mountain and the Iron Hills. (ref) After Smaug descended upon the Lonely Mountain, we catch up to the events of The Hobbit.
The Lonely Mountain is also known as Erebor (Sindarin for … Lonely Mountain). The connection between the realms can be clearly in the door design for Erebor and Skarháld