Inspire second quartet visiting gallery (8)
I believe the answer is:
motivate
'inspire' is the definition.
(thesaurus)
'second quartet visiting gallery' is the wordplay.
'second' becomes 'mo' (both are short periods of time).
'quartet' becomes 'IV' (Roman numerals for four).
'visiting' indicates putting letters inside.
'gallery' becomes 'tate' (Tate Modern, Tate Britain etc).
'iv' put into 'tate' is 'tivate'.
'mo'+'tivate'='MOTIVATE'
(Other definitions for motivate that I've seen before include "Stimulate to action" , "Stimulate the interest of" , "Give an incentive for action" , "Prompt" , "Induce" .)