The left-hander, ranked No 117, will face the winner of the match between American Madison Keys, the top seed, and Dane Clara Tauson, the No 5 seed, for a place in the final.
Montgomery beat Japanese lucky loser Mai Hontama (6-3, 2-6, 6-0) and Japanese qualifier Nao Hibino (6-2, 6-2) in the previous rounds of the tournament.
Earlier in the week, Pera, ranked No 71, won against Andorran qualifier Victoria Jimenez Kasintseva (6-2, 7-5) and Canadian Rebecca Marino (7-6 (2), 6-4).
Auckland WTA 250, other quarter-finals matches (ASB Tennis Arena, hard, USD 275,094):
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Four-time Grand Slam champion pulls out of her first WTA final in three years after winning the first set of the Auckland Open.Naomi Osaka has retired from the
Naomi Osaka of Japan reacts prior to withdrawing from her match against Clara Tauson of Denmark ... [+] during their women's singles final match at the WTA Auck
Naomi Osaka has suffered an injury scare ahead of the Australian Open later this month after retiring from the final of the ASB Cl