The 19-year-old will move to Croft Park in a work experience arrangement.
Quinn rose to prominence last season when he hit a hat-trick in the Magpies' FA Youth Cup third-round win over Sunderland at St James' Park.
He was also part of the matchday squad for Newcastle United's pre-season fixture against Sheffield Wednesday.
However, the youngster has struggled to break into Peter Beardsley's Under-21 side on a regular basis so far this term.
Quinn could make his competitive debut for Bltyh Spartans in the FA Cup second qualifying-round tie at Skelmersdale United on Saturday.