Tourney Grounds and Forgotten Temple,while Active, give QI boosts.
Yggdrasil, Shinto Temple, Royal Bathhouse, Sun Temple, Mughal Temple and the Stage of the Sacred Instruments at max level - and ONLY at max level - also give QI Boosts.
Decorations in your QI settlement provide QI boosts.
That's all - currently nothing else gives QI boosts.