Will my progress reset?

Yes. Quest progress is tied to the current 24-hour cycle. When the daily timer ends and the quests reset, any unfinished progress from the previous day is cleared, and you will start fresh with the new set.
 

Example: If your quest is “Win 5 rounds” and you have only won 3 by the time the 24-hour period ends, the quest will reset, and you will start from 0 on the next day’s quests.