The host asked the couple to name their first-born son after him.
The Italian celebrity chef, 44, fronts the reboot of the well-loved series, which originally ran from 1980 to 2002.
In scenes due to air on Sunday, contestant Liam cuts off D’Acampo as he asks a question, saying: “I appreciate the question Gino, but I’ve got a question to ask myself.”
The contestant then leads his girlfriend, Nadine, to the centre of the studio stage.
Both of them now crying, he says: “I love you so much. You’ve been in my life for four years now and truly I have never loved someone more in my whole life.
“The minute I set my eyes on you, I thought you were the most amazing girl I have ever met in my entire life.”
Liam then drops to one knee and presents a ring from his pocket, with Nadine saying yes.
Visibly shocked, D’Acampo requests that if their first child is a boy they name it Gino after him.
The new series of Family Fortunes has retained the classic format, with families playing for the chance to win “big money” by guessing the most popular replies to survey-based questions posed to 100 people.
Based on the US game show Family Feud, the Family Fortunes format has aired in more than 67 territories worldwide over the past 44 years.
Family Fortunes With Gino D’Acampo is on Sunday at 8pm on ITV.