No, once you register for a quest, that quest will be accessible in your “My Account” area under the "Registered Quests" heading.
If you have registered for an Adventure Quest, then the quest will be available to begin immediately, although you don’t have to start it immediately. The quest will remain active and will be displayed in your “My Account” area until it is started and ultimately completed.
If you have registered for a Treasure Quest, the quest will be displayed in your “My Account” area as well. Treasure Quests, however, will start at the same time for all registered participants, and you will not be able to begin until the set date and time of the quest after which you may start. Since the first participant to complete the Treasure Quest wins the prize, it is to your advantage to begin the quest as soon as it has started.