I was trying to look at your lodestone to see if you had progressed MSQ enough in order to unlock it but unless you have a different character your thaumaturge is only 11. Also as far as I could see it looks like you're still in Heavensward content as well which until you gain access to Stormblood MSQ you cant get those quests either.

That all being said... a quick google search for ffxiv black mage 70 quest gave me the top result as https://ffxiv.consolegameswiki.com/w...ck_mage_quests , which in turn leads to the quest One Golem to Rule Them All, which is from Lalai in Ul'dah, Steps of Thal at x12.9 y13.3. If you do in fact have a black mage unlocked and at level 70 and this quest doesnt show up check your journal to see what the last completed black mage quest was and work from there, otherwise at best all I can say is make sure you have your Job Stone equipped and make sure your MSQ has progressed enough to actually be able to gain access to the quest.