You can buy Pickaxes from Silent Rooster (K-10) in the Morimar Basalt Fields for 200 gil each. He is a few steps away from Apolliane. You will likely need many Pickaxes, but you can only attempt to excavate 25 times at once. You then can either zone out to reset, or wait some amount of time due to fatigue. Depending on your method, you may want multiple stacks of pickaxes.
Known Mining Points include J-10, J-11, I-9, H-5, J-6, J-7, I-7, F-10, and F-8.
You cannot trade her a Marble Nugget that you've obtained any other way besides mining it yourself.
Hardened magma forms crags that dot the landscape, hindering travel. Collect one chunk of marble nugget and bring it back to Apolliane in exchange for her teaching you the arts of destroying these obstacles.