The Reds have made a perfect start to the new Premier League campaign with four successive victories, and also claimed the UEFA Super Cup in Istanbul.
Their taste of glory in Madrid in June, and the scenes that followed during the parade in the city, is an obvious motivation to secure more silverware.
And Wijnaldum feels Liverpool can set about achieving that objective in an assured way.
“Winning the Champions League has given us all peace of mind,” said the Dutchman.
“Now we know what it’s like to win the biggest trophy of them all – and we just want to keep going from here and add even more titles.
“But Jürgen Klopp is not putting extreme pressure on us to win the Premier League. He does not tell us, ‘Now you must win this or that trophy’. He is very relaxed.
“He did not do that during the past season, either. His idea is that we get the maximum out of every individual every single day that we are together. And then we will see where that brings us at the end of the season.
“Just look at the list of world-class players who have not won the Champions League during their careers. It contains more famous names than you would think – and it tells you that it is really special we have won it with Liverpool.”
Liverpool’s defence of the Champions League begins this month after they were pitted against Napoli, Salzburg and Genk in the group stage.
UEFA’s annual awards were also presented at the draw in Monaco and Virgil van Dijk was announced as the Men’s Player of the Year for 2018-19 in recognition of his impeccable form for club and country.
As his teammate for the Reds and the Netherlands, the news did not surprise Wijnaldum.
“I was not surprised when Virgil won the UEFA best player award ahead of Lionel Messi and Cristiano Ronaldo. I actually expected him to win it,” added the midfielder.
“Virgil has had a blinder of a season and has been so successful. Nothing about him surprises me anymore.”