Well, the initial quest, "The Weaponsmith of Legend", begins at Vesper Bay, though it sends you to the North Shroud. After that, you can do "A Relic Reborn" which begins in the North Shroud. It's followed by a series of additional quests (also starting in the North Shroud) for upgrading your relic, until it reaches its final, Zeta, form.

("The Weaponsmith of Legend" is a one time quest. If want to get relic weapons for multiple jobs, all subsequent ones begin with "A Relic Reborn" which is repeatable once per job.)

Console Games Wiki has a good rundown of the whole process here:
https://ffxiv.consolegameswiki.com/w..._Weapons/Quest
The Relic Weapons and Zodiac Weapons sections are the ones for level 50. Anima Weapons and Anemos Gear are for Heavensward and Stormblood, respectively.