Can I set the "Exercise" goal for my Apple Watch Activity app?
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").