There are a few different types of quests that include items. The two main item quests require you to either Have or Place an item or a series of items in your garden. Please remember that Have and Place are very different requirements.
Have: You need to have this item in your garden. You will receive credit if you purchased the item before or after the quest appeared.
Place: You have to purchase and place the item in the garden after the quest appears. Any items you purchased before DO NOT count toward finishing the quest.