We have to agree to differ on this one. As I have said before, I was really, really down for a long time when my son was little and at his most hyperactive. (I had never been depressed like that before. I'd usually been the lucky one who got whatever she wanted, travelled the world, husband that I adored etc. etc.) It was my son's teacher, Mary, a wonderful person for whom I had the greatest respect, who hauled me back up again. So much depends on the language you use. In those days, if my mum or someone who knew me well had said to Mary simply that I was "a bit down" I would not have considered that as a betrayal of trust at all. On the contrary, it might have meant that I got the help which I desperately needed a whole lot quicker, and my son would have got his happy mum back again too. So much depends on the individuals concerned, this forum is about sharing ideas. It's a bit like the "bringing up baby" books, you can read them all and see which one you like the most!
I agree with Ladybird not to go behind your daughter's back to discuss her situation with the teacher. Fine if your granddaughter is picking up on the sadness, your daughter agrees and wants the teacher to be aware of why, but not feel able to say so herself.
if the doctor says it is definately not depression, perhaps you could go with your daughter and ask for counselling as talking to a non judgemental stranger can help.