Initially noting information, nerds extremely square (4)
I believe the answer is:
nine
'square' is the definition.
(nine is a square number)
'initially noting information nerds extremely' is the wordplay.
'initially' suggests taking the first letters.
The initial letters of 'noting information nerds extremely' is 'NINE'.
(Other definitions for nine that I've seen before include "3x3" , "Number of feline existences" , "Number of lepers who failed to return" , "A number of worthies" , "A stitch in time saves them" .)