Via the lottery system, you will eventually win free move building tokens. To know if you currently have one, click your storehouse and scroll all the way down, if you see move building in the list, you can do it.

To move the building, make sure it's not level 20 but level 19 (level 20 will need 2 vouchers, level 19 needs only one), select another mountain in your city where you want to place the new quarry, click it and scroll down, you will see the quarry and a gold cost, click to pay, it will pop a new window asking how you want to pay for the move, you can select either gold or the free vouchers and then, you have the building in the new spot.

Also remember, armies only have 4 options when they move from one tile to another, they can only go front, back, left or right, nothing else that means only 2 walls at each entrance (or one wall and one gate) instead of three walls (or two walls and one gate).