The Hammers, who currently sit 10th in the Women's Super League, progressed as Group D leaders, while Championship high-flyers Durham finished second in Group A.
The other last four tie pits Bristol City against Leicester City, with the fixtures due to take place on the 3rd and 4th February.
City progressed to the knockout stages of the Continental Cup by finishing top of Group C.
They take on the current holders, Chelsea, this Wednesday at the Academy Stadium.
Kick-off is at 17:30pm (UK) and you can follow all of the quarter-final action as it unfolds on CITY+.