Elsewhere in the draw, Inverness CT host Raith Rovers while there’s an all-Championship tie at Palmerston Park as Queen of the South welcome Falkirk.
The nature of the draw means at least two teams from the lower leagues will reach the semi finals.
Spartans will have to travel south for their rematch with Berwick Rangers, after Ally MacKinnon’s last-gasp equaliser forced a replace.
The tie will be replayed at Shielfield Park on February 17th.
The quarter final matches will be played on the weekend of March 7 and 8.
Draw in full
Dundee United v Celtic
Hibernian v Spartans / Berwick Rangers
Inverness CT v Raith Rovers
Queen of the South v Falkirk