Messi has four winner's medals to his name since 2005, the most recent of which came in 2015 when helping Barca to a 3-1 win over Juventus in Berlin.
The Catalan giants have been knocked out in the quarters three times since then, however, with arch-rivals Madrid lifting the famous trophy in each of those campaigns.
Suarez believes that the true barometer of success for any player is the number of Champions League titles won, making success this time around all the more important.
"The Champions League is the competition which defines you, as the best players and teams of Europe are in it," he told RAC1. "There's an extra motivation to never relax. Our desire, motivation and expectations are for this to be a great year.
"It's obviously an irritation when your rival has won the past three Champions Leagues, as it's historic. It gives you even more desire to turn the situation around.
"[Messi], as a player, deserves many more Champions League titles than he has. He is full of desire after the World Cup and after what happened to us in the Champions League last year. He has always taken on a leadership role and is keen to lift some trophies."
Barcelona have been drawn in a group with Tottenham Hotspur, Inter Milan and PSV Eindhoven for the 2018-19 edition of the competition.