I noticed you did say you don't know if maybe you already done the quest or not. Easy way to check is open you're armory chest. (CTRL+I). Look in the bottom box on the right. That is where all soul stones are keep for any Main job you have unlocked. It you see the summoner soul stone there then that means you did do the quest. All you need to do now is equip it and you're a summoner. Good luck and hope that helps.