Messi had already left PSG when Kvaratskhelia joined last January.
However, the Inter Miami was keen to meet the Georgian at last year's Ballon d'Or ceremony.
Kvaratskhelia recalled: "He was sitting to my right.
"All of a sudden, Leo stood up, and I had no idea why or where he was going. We were asked to remain seated during breaks, but he walked straight toward me."
Taken by surprise, Kvaratskhelia didn't know how to react: "At one point he held out his hand to me. I was speechless and immediately got up to shake it. After that gesture, he nodded to me and returned to his seat."
A simple gesture, yet one full of meaning for the former Napoli star: "That day, he earned all my respect. For a player like him to stand up and come say hello to me… it was incredible.
"Messi is an extraordinary person, whom I respect and appreciate enormously."