No, not the quest. For example, the female hairstyle notes are obtained from Giselle in Dewata and for every note you have to get 100 of whatever item she requests. Each hairstyle note only gives you the next hairstyle so for me to want to use hairstyle #99, I'd have to collect 12 hairstyle notes since there are only 87 default ones. And 12 hairstyle notes means 1200 items farmed and for me to get the same hairstyle on characters across 5 different accounts, I'd have to farm 6000 items.
Like the CNY quest costumes that can be shared across via accounts with the vending machine, would be good to have hairstyles shared too.