Robertson, who won the trophy with Liverpool in 2019, believes his Scotland teammate Gilmour won't let the success go to his head.
"It only changes you if you allow it to change you," he told the Guardian.
"I have spoken to Billy a couple of times and the wee man seems very hungry to achieve everything he can
"To be able to get a Champions League winner's medal so early is incredible but I don't believe that will be him finished.
"When you get that feeling, when you get the night out that follows, the celebrations with your family and your fans, that makes you hungry for more and I'm sure Billy will be the exact same.
"He has a huge career ahead of him and it has already started so well. I have no doubt that he will stay humble, hungry and go on to achieve a lot more."