The Apple Watch Activity allows for setting an overall "Move" goal, but I don't see how to change the green "Exercise" goal from the default 30 minutes. Is there a way to accomplish this?
No there is no way to accomplish this.
From Apple - Apple Watch
You close the Exercise ring when you reach the globally recommended 30 minutes of exercise per day. And you don’t even have to do it all at once, which means you’re more likely to fit it into your day.
From all searches I have performed I have not found any possible way to change it and the above quote suggests that 30 minutes is the amount Apple has fixed on (the call it "recommended").
You aren't limited to one time around the ring in the Activity app. You get a double arrow for 1 hour of exercise, for example. You also receive awards for doubling or tripling the exercise ring.
You can use the Workout app to set whatever exercise goal you like. And you get a custom ring to fill in that app, based on target calories, distance, or time.
watchOS 7 enables changing the green Exercise Goal.
Apple Watch > Activity app > Scroll to Bottom > Change Goals