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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