Here’s one I did earlier:

In iD (the default editor on the website), scroll down to the “tags” area and add a new tag “designation” with a value of “quiet_lane”.

You’d need to know that the data that you were adding was compatible with OpenStreetMap’s licence (which of course it will be if you’ve recorded where the quiet lanes are yourself). If you want to add the data from quietlanessuffolk.co.uk and it’s not otherwise clear, you could ask them whether its OK to add the data to OpenStreetMap.