I just checked every quest in: North Qeynos, South Qeynos, Qeynos Hills, Surefall Glade, Halas, West Karana and North Karana, and I can confirm that there are exactly zero low-level Bard quests for a magic weapon. There are low-level magic weapon quests for Warriors, Paladins, Clerics, Shaman, and Rogues, but not Bards.
There is
one quest for a magic weapon for Bards, but it requires beating the boss of Blackburrow, so unless you can find a high-level player to help you with that you'll have to go with Dino's suggestion, or some other dropped item.
Or of course you could also try to find someone to port you to EC, where you can likely beg a magic weapon off someone.