I've ended (cashed-in) quests with item/weapon rewards while I had a full inventory, and I couldn't tell what happened to the items. I don't think they made it to my inventory, but they weren't on the floor and there wasn't an option to close-out the quest again. Were the items liquidated into money, held on the side somewhere, or did they just disappear? What happens in this case?
When you have a full inventory, any quest rewards you get should still end up in your backpack. For example, if you have 15/15 backpack slots full and you get a SMG from a quest, your gun will be in your inventory and your backpack should display 16/15 capacity now.