You will gain access to the Quarry upon completion of the Crafts Room in the Community Center which rewards you with the Bridge Repair. The bridge is located east of The Mines and the Adventurer's Guild. If the Community Center was sold to the Joja Corporation and turned into the Joja Warehouse you will instead acquire the Bridge Repair and subsequent access to the Quarry once all Crafts Room Bundles have been purchased from the Joja Community Development Form for 25,000g.
The Quarry itself can be home to every Ore Deposit in the game from Copper Ore to Iridium Ore, however Iridium Ore deposits are much rarer and you may have to wait several months for new ones to spawn in. Additionally you can also find Geodes ranging from Omni Geodes to Magma Geodes.
Each day a selection of new items will spawn, so to prevent valuable spawns from being blocked by existing rocks, make sure to keep the quarry entirely cleared.