I seriously think weapon quests are very few and far between. I just don't see any quests anywhere at any level that provides a decent item. Well except the heritage quests like PGT for scouts but that is much later on. Most quests seem to be coin/xp, armor, food/drink, or access.
My recommendation would be to check the brokers for something to purchase or to use the new /lfw search tool and find a crafter who will make you something.