1. Make sure your backpack isn't full.
2. Make sure your preferences aren't set to dispose of anything on drop
3. Make sure you're not accidentally hitting the "X" when they drop
4. *If you're still having issues getting a drop, head to the buff market, and get the buff "Find Item." If you can't afford the cheapest one; ask a fellow player (the community is very helpful and welcoming to new players). Specifically in Global Chat below your map.
Items do have drops rates and you could just be getting un-lucky. Step 4 will help with that, if that is the case but, you shouldn't be experiencing drop issues for starter quests. Just make sure you have room in your backpack, and none of your preferences have been tampered with. They will drop.