If you are able to complete Quests within the specified time, additional rewards will be awarded.
The time limit varies from Quest to Quest.
You can confirm the time limit by checking the gauge in the top-right corner of the Quest screen. If the Quest is completed before the gold portion of the gauge has reached the treasure chest icon, you will be eligible to receive time bonus rewards.
Obtained time bonus rewards will be displayed on the Quest reward screen after you have completed the Quest.