As to how to implement it - progressing quests would be one way to do it, at least some impactful ones, like the one in the Druid Grove.
I generally dislike timers - what it means is that we have to do resource management with an unknown source. Do you clearly communicate timer for each times quest? That's a bit heavy handed. If you don't then that become a mess as the player doesn't know how they could manage their rests - they get encouraged not to rest, and then they don't use abilities, and then game becomes no fun. If resting is limited, then players have an idea on how they are doing.
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Yes, I personally think that it may be a bad decision to force timers on everyone. Not to mention it would cost dev team time and resources to implement.
That said, I would appreciate if there was a timer option, either on high difficulties, or a toggle (similar to DOS2 Honour mode) that would be great.
How to communicate the timer - that is difficult. The easiest way would be through the log, but as you mentioned it may be a tad heavy handed
