The quality of a bore? (7)
I believe the answer is:
calibre
I believe this clue is a double definition.
'the quality' is the first definition.
('calibre' can be a synonym of 'quality')
'a bore?' is the second definition.
('calibre' can be a synonym of 'bore')
'of' is the link.
(Other definitions for calibre that I've seen before include "Quality - gun barrel bore" , "Level of ability; bore" , "gifts" , "Diameter of a gun barrel" , "Diameter of bullet or shell" .)