Including templates in the quick compose window complements the more practical use of "phone call" templates if we ever decided to add a quick log message window. Assuming quick log message copied quick compose they would both ideally support templates.
See the
CHD-879 comments for an explanation on quick log message templates.
Ideally quick log message would support templates for phone call logging, which was the original source of inspiration for this particular request.
That way a customer calls -> you search for his name -> click quick log message -> open up the call template -> fill in the details -> save.