For the quest "The Veteran of the South Gate" it requires you to speak with Duvane. I did so, however when I try to hit complete, it states "There is no empty slot". It won't allow me  to select and complete any other quests afterwards either. Am I just missing something simple? 

That usually means your inventory is full. The quest has a reward to give you, but if you dont have an open slot in your inventory it will not let you complete the quest. Just sell some junk loot and you should be good to go.

Sell or throw away some junk and you will be able to accept the rewards.