Marble is used in making Masterpieces.
To obtain marble unlock the Bailiff office and assign a Bailiff. The Bailiff is able to have 2 types of Mandates: courting or prospecting. Choose the prospecting action. There is a chance your Bailiff will find Marble.
If your Bailiff finds a resource it's deposit will become visible. Purchase the land the deposit resides on. You can then click on the deposit and Edit it like you would a Monument. Add a Quarry and assign a worker.