How am I supposed to turn in my relic weapon quest?

[Kirika-]

Nope, the weapon had to be in your inventory for every single relic since the start of ARR. You have just remembered wrong.

Some of the progression steps require the relic be equipped in order to proceed (mainly the steps were a quest check is made to determine if you actually qualify to get the quest), however any step that resulted in any change to the relic required it to be in your inventory bag.
As in, the steps that resulted in the item having a new name or different stats/appearances, means the existing version of the item is "destroyed" and "replaced" with the new version, of with can't be preformed if the item is equipped because during that destroy/replace stage, you can not exist without a job specific main hand item equipped.
During those steps, you need to have a different job specific main hand equipped in-order to be able to de-equip your current relic, but still be in the correct quest job.

That said, it is somewhat of a pain to remember to go get a cheap-o weapon before going to a zone with no cheap-o weapon vendors for the current set of relic quests.....
Rowena, you are missing a chance at profit.
Set one of your girls up in Gangos with a line of junk weapons (sold at an extreme profit ... $$$$$$$$) so the Gil-bloated adventurer can progress on their relic quest more seamlessly