I tested and verified, if you do this quest and do not have all of your artifacts for the class, then go get the one(s) you are missing, the new one will also be fully empowered. You may just have to stop by the artifact forge in your class hall.
I copied and pasted this quest list from WOWHead comments so all credit goes to the person that posted the comment (Bursi):
By Bursi (27,897 – 5·34·312) 9 hr 23 min ago (Patch 7.3.5)
First quest might only pop up after killing Argus once on that account. Relog should help, if it does not show up for you or close your client completely, if you have done step 9 and The Speaker’s Call does not trigger (as described here).
You can switch the zone phasing at Zidormi, if you can’t see your quest giver in Silithus.
If you just want to empower your artifacts quickly, you can skip the optional quests.
Summons to Stormwind - go to Stormwind and meet Master Mathias Shaw Gifts of the Fallen - click his dialogue (Video) Witness to the Wound - travel to Silithus. You can use the portal to Uldum in Stormwind (/way Stormwind 75.10 20.36) Free Samples - kill Bilgewater Laborer and collect 8 ores It's a Sabotage - place 10 Dynamite Bundle (optional) Desert Research - turn in at Tammy Tinkspinner near you (optional) The Source of Power - collect 10 brains (optional) Larvae By The Dozen - click 12 broodlings The Twilight Survivor - kill Michiel Voidstrider and (optional) accept/complete A Wee Bit O' Cloth at the toilet A Recent Arrival - speak to Archmage Khadgar The Speaker's Perspective - go to Magni Bronzebeard in Silithus The Blood of Azeroth - return to Staghelm Point (Quest #50056) - speak to Magni Bronzebeard, she'll transport you to Magni The Power in Our Hands - click the dialogue (Video)
Questline (incl. optional quests) took around half an hour, started from Sw/Uldum. Without optionals and cinematics it were 15 min.
Now all of your character’s artifact weapons are empowered to maximum.