We're not told, precisely, that John the Baptizer's father, Zechariah, actually repented. But he was muted, unable to talk, after he failed to believe God's word through the messenger sent to him, when he was doing priestly service for Israel, in the temple.
God's graciously blessed him and his wife Elizabeth, in their old age, with the birth of a son, the forerunner of the Christ, John the Baptizer. I'm sure Zechariah repented of his unbelief along the way. At John's birth his muted state was removed, as he confirmed the name that had been given to him from the messenger/angel, and then we have the Benedictus as Zechariah bursts in thoughtful praise to God.
Repentance is something we do, whether we feel like it or not. We agree with God and God's word, that we are wrong in word or deed, including our inner world of thoughts and attitudes. This is a humbling of ourselves before God. And God will answer in his grace. God changes us, and gives us, in time, a sense of his peace.
John the Baptizer was sent to prepare the way for the Lord. He is important as well, as we think of Advent. Our hearts need to be prepared to receive, in Jesus, God's coming into this world- all of it, including every part of our existence.