Right next to the Alhambra grounds we stumbled into a huge garden (seemed like it was at least 20 acres, maybe more) called Carmen de los Martires Jardin Monumental Romantico (loosely translated, Carmen of the Martyrs Monumental Romantic Garden).
These gardens were spectacular. Peacocks roamed around, there were fountains, grottos, and statues all over, and the best part was we didn't see another soul for a while - it really felt like a secret garden to us (in fact, for a while we weren't even sure we were allowed to be there). Eventually we saw a maintenance man with a leaf blower and a girl reading on a bench along one of the many paths, but it didn't feel like these gardens were frequented very often (but who knows?), and there was no mention of them in the Rick Steves' Spain 2009 edition we had.
I mention them because you were asking about finding a quiet spot and for suggestions in the area, and we thought they were fantastic. Plus, if you are going to be there during a hot day, these gardens had plenty of shade to cool things down. HERE is a link to some info I found on them through Google. In fact, I liked these gardens more than the Generalife at the Alhambra, but it was the peacocks that helped sway my vote.