Apologies but I cannot answer your question directly, but I can tell you what I do know.
I noticed whilst doing story quests that I was getting around 4k xp for a 5mins quest..,,and it takes 300xp to lvl as a lvl1.
So I picked up say like a lvl16 story quest as Paladin, did the quest and attempted to hand it in
as a lvl1 Arcanist. I was interested to see that a red cross appeared above the NPC when i changed jobs, so I could not hand it in.
This suggests they "lock" the quest to the job that the quest was started with.
I think it's simpler than that. There is a level requirement to finish
a quest as well as a level requirement to activate
the quest. I've noticed during both Phase 3 and Phase 4 that when I traveled to Gridania to unlock and level CNJ that there were certain quests in the Central Shroud that I could activate before level 10 (I was level 8 at the time, I believe). But when I completed the quest objectives, I saw the red cross on the NPC like you saw. I was CNJ the whole time. When I finally hit level 10, the red cross changed to the usual quest completed icon and I was able to finish the quest.
I'm betting that the quest you were trying to complete has a minimum level requirement for finishing, just like the quests I was doing. That would explain why you couldn't finish it as a lvl 1 class.
I'm not sure why some quests would have a higher minimum level to finish the quest than it requires for flagging the quest. I wonder if it's a bug, and that the two level requirements are supposed to be the same (which would make more sense).