Quite Places in Edinburgh?

you can walk down to the Stockbridge area from York place and access the water of Leith walkway from there ,but I would take a bus out to Slatford(Visitor centre) or Roseburn and walk from there to Stockbridge, done lots of the water of Leith walks from balerno to Leith in one stretch or in sections.
Crammond is quite quiet ,I will be down their on Saturday and going out to the island to do some berry picking.(nice little pub in Crammond)

If you like castles, consider taking the bus to Craigmiller castle. The bus stops about a quarter mile from the castle. You could also take the bus out to Roslin Chapel. Both places are away from the crowds.

My wife and I did not hike it when were there but the hike up to Arthur's Seat looked interesting.

I would walk up King Arthur's Seat if you are physically able to do so. There are many different routes up it. My son and I went up the stairs carved out of stone that start between the two large hills. It looks worse than it is from the ground but still challenging. There are no public bathrooms around there except for the ones across the open field east of the Scottish Parliament. Walk to the Cambridge Pub for a nice pint and a good burger too.

Sorry but what's a "quite place"?

probably a typo and they meant quiet.

The most wonderful place we visited in Edinburgh was the Water of Leith park.

https://www.edinburghguide.com/parks/waterofleith

The Royal Botanic Garden, at Inverleith on the north side.
They are free to enter, and for an extra charge you can go inside the huge Victorian glasshouse.
There is a great gift shop, a good café in a building out in the grounds, and a newish restaurant.
Lots of information throughout the grounds, and you could easily spend a whole day there.
You can get a pretty good view of the Castle from there too.
Good in any weather!
Plenty buses stop there.

http://www.rbge.org.uk/