There's a small room past the concierge desk on the left, right near where the DVC desk is. It's kind of like a study. I was in there one day for several hours, and saw only 2 or 3 people come in to check out the balcony. You could lay him out there, and there are chairs for you that you can turn around and look out the full-length glass windows at the savannah.
That's the Sunset Lounge - and there are various activities scheduled in there at various times of the day ... so check the schedule before you plan on spending some "quiet time" there ...

You could also just grab a lounge chair by the pool ...
And there are small overlooks off the hallways on each side of the "big window" that overlook the Arusha Savannah - they may be quiet, too ...