Central reached the last eight with a 4-0 win over North Shore United in the previous round, while Wairarapa edged past Miramar Rangers 2-1.
In Wellington, 2006 Cup winners Western Suburbs host 2010 runners-up Bay Olympic.
The two teams experienced contrasting paths, with Bay Olympic prevailing over Western Springs 3-2, while Western Suburbs brushed past Stop Out 5-1.
The remaining quarter-final is an all-South Island affair between Nelson Suburbs and Cashmere Technical.
Cashmere, who won back-to-back Chatham Cup titles in 2013 and 2014, will be prohibitive favourites, having already beaten Nelson four times this year.
The quarter-finals will be played on the weekend of August 5-6, with the details to be confirmed in due course by the clubs and federations.
ISPS Handa Chatham Cup Quarter-Finals
(August 5-6, match details TBC)
Central United v Wairarapa United
Western Suburbs v Bay Olympic
Birkenhead United v Onehunga Sports
Nelson Suburbs v Cashmere Technical