In Scrolls, it seems like Trials can only be completed once for gold. Do they reset after some time for the ability to earn additional gold, a la daily quests in WoW, or once I finish them is my only option to earn by playing Quick Match?
No, they don't reset. Although this may change in the future, when you first download and play the game, I recall reading a specific message that states that upon the first, initial completion of a Trial, you are given a reward. After that, you can continue to play the trials but you will not receive rewards for winning the trials.
If you would like to earn additional gold, just play Quick Matches or player-to-player games.