Is that the one where it gives you a present? I think you just need to keep doing it a bunch. Eventually, it'll happen. You might take a look at the skills tree and see if there's one there that'll help you be the trees' bees' knees.
I have three trees in my backyard: two bubble trees and a wood tree. I just spent an entire hour watering and petting the bubble trees every five minutes (basically as soon as the menu option became available) and the quest is still not fulfilled. The trees did not change in appearance at all. I refrained from harvesting just in case that would slow things down.
I am seriously starting to think the quest is just broken.
Growing a tree takes time, not just water and petting. I think they have hidden time and health bars just like the visible ones for crops: one represents the maturity of the tree, and the other represents its happiness/health. When the tree is fully matured by age, and then you pet/water it to full happiness/health, the quest seems to trigger. That's how it was with me and the tree I planted in my hard for this exact purpose. No idea if this is true, but anyway, I hope it helps!
I happened upon a street that had two very small (definitely not fully grown) cherry trees. I watered/pet them both and after doing so for the 2nd one, I had completed the quest.
I planted 4 trees in my new house. I have since raised them to full trees. I've probably pet/watered each one about 30-40 times. I have used 15 charges of fertilidust on a single tree.