You can open your journal (in the 'Duty' menu) to see the active quests. You will see a class/job icon in the quest name. If you want to see the quests that you already did you can click in the 'Complete' button and select 'Class & Job Quests'.

Here is the list of the conjurer quests:
https://na.finalfantasyxiv.com/lodes...7&category3=41

And here for White Mage:
https://na.finalfantasyxiv.com/lodes...7&category3=45

You can identify the MSQ by its icon.
https://ffxiv.gamerescape.com/wiki/D...ns#Quest_Types