Of course they do - but, of course, how you define "monergism" would be determinative. Did Mary cooperate with this grace? You betcha. Did she choose to? Absolutely. Did she have a free choice? Without a doubt. Could it have come out any other way? Absolutely not.
Interesting take on the meaning of Luke 1:38. The Fathers generally pointed out Panagia's "assent" or "obedience", both terms in Greek implying a free choice, not the act of an automoton and therefore, while her assent freely given was foreseen, it was not inevitable. For example:
"And when she asked in her perplexity, How can this be, seeing I know not a man ? the angel again answered her, The Holy Spirit shall came upon thee, and the power of the Highest shall overshadow thee. Therefore also that holy thing which shall be born of thee shall be called the Son of God . And she said to him, Behold the handmaid of the Lord: be it unto me according to Thy word . So then, after the assent of the holy Virgin, the Holy Spirit descended on her, according to the word of the Lord which the angel spoke, purifying her , and granting her power to receive the divinity of the Word, and likewise power to bring forth." +John of Damascus, An Exact Exposition of the Orthodox Faith, Book III, Chapter 2
"For inasmuch as He had a pre-existence as a saving Being, it was necessary that what might be saved should also be called into existence, in order that the Being who saves should not exist in vain. In accordance with this design, Mary the Virgin is found obedient, saying, “Behold the handmaid of the Lord; be it unto me according to thy word.” But Eve was disobedient; for she did not obey when as yet she was a virgin. And even as she, having indeed a husband, Adam, but being nevertheless as yet a virgin (for in Paradise “they were both naked, and were not ashamed,” inasmuch as they, having been created a short time previously, had no understanding of the procreation of children: for it was necessary that they should first come to adult age, and then multiply from that time onward), having become disobedient, was made the cause of death, both to herself and to the entire human race; so also did Mary, having a man betrothed [to her], and being nevertheless a virgin, by yielding obedience, become the cause of salvation, both to herself and the whole human race." +Irenaeus Against Heresies, Book III, Chapter 22
"Wherefore we acknowledge one Christ, one Son, our Lord Jesus Christ; knowing that being coeternal with His own Father as touching His Godhead, by virtue of which also He is creator of all, He deigned, after the assent of the Holy Virgin, when she said to the angel ‘Behold the handmaid of the Lord, be it unto me according to thy word’ to build after an ineffable fashion a temple out of her for Himself, and to unite this temple to Himself by her conception, not taking and uniting with Himself a body coeternal with His own substance, and brought from heaven, but of the matter of our substance, that is of the Virgin." Blessed Theodoretus, Bishop of Cyrus, Dialogue II
In consequence of this, Orthodoxy also teaches and holds, and I suspect the Latin Church does too, "...the Virign Mary is honored noy only because God chose her, or because she bore the Son of God in the flesh, but also because she herself chose to believe and obey God firmly."