No doubt she did, both out of personal desire for a child and "to take away my reproach among men."
"the conception was solely through divine grace
Unless I've totally misunderstood everything, the conception of John was through natural sex. The conception of Jesus by Mary was "solely through divine grace," no human father nor natural reproductive act involved.
Righteous Elizabeth the mother of St John the Baptist...
The Righteous Elizabeth was the mother of the holy Prophet, Forerunner and Baptist of the Lord, John. She was descended from the lineage of Aaron, and St Elizabeth was the sister of St Anna, the mother of the Most Holy Theotokos. The righteous spouses, “walking in all the commandments of the Lord (Luke 1:6), suffered barrenness, which in those times was considered a punishment from God.
When Elizabeth gave birth to a son, through the inspiration of the Holy Spirit she announced that his name was John, although no one in their family had this name.
They asked Zachariah (who had been rendered mute) what the child’s name was, and he wrote the name John down on a tablet. Immediately the gift of speech returned to him, and inspired by the Holy Spirit, he began to prophesy about his son as the Forerunner of the Lord.
When King Herod heard from the Magi about the birth of the Messiah, he decided to kill all the infants up to two years old at Bethlehem and the surrounding area, hoping that the new-born Messiah would be among them.
Herod knew about John’s unusual birth and he wanted to kill him, fearing that he was the foretold King of the Jews. But Elizabeth hid herself and the infant in the hills. The murderers searched everywhere for John. Elizabeth, when she saw her pursuers, began to implore God for their safety, and immediately the hill opened up and concealed her and the infant from their pursuers.
In these tragic days St Zachariah was taking his turn at the services in the Temple. Soldiers sent by Herod tried in vain to learn from him the whereabouts of his son. Then, by command of Herod, they murdered this holy prophet, having stabbed him between the temple and the altar (MT 23: 35). Elizabeth died forty days after her husband, and St John, preserved by the Lord, dwelt in the wilderness until the day of his appearance to the nation of Israel.
On the Greek calendar, Sts Zachariah and Elizabeth are also commemorated on June 24, the Feast of the Nativity of St John the Baptist.
https://oca.org/saints/lives/2016/09/05/102503-righteous-elizabeth-the-mother-of-st-john-the-baptist
Natural sex wasn't working. That's why they needed divine intervention in their old age - implying years of trying to conceive. So the divine intervention was to allow the natural sex to work.