The Mexico captain was presented by the Serie A club on Friday after a deal was struck with Leon to take him to Serie A.
Centre-back Marquez has previously played for Monaco and Barcelona but his transfer to Verona will be his first taste of the Italian top flight, and there is one particular battle the 35-year-old is looking forward to.
"I want to play against Balotelli," said Marquez. "He is a player with unique qualities who I still haven't faced."
And the experienced defender, who played in his fourth World Cup in Brazil, hopes he can help Verona improve on their impressive campaign last year.
Throughout the season a place in the UEFA Europa League looked a possibility until Verona fell away and had to settle for a mid-table finish.
Marquez is aiming higher this time around, but concedes securing their place in Serie A is the first priority.
"I haven't come here just to achieve safety but to do something more," he added. "I don't know whether we will win the Scudetto but we will always aim for the maximum.
"The first goal is survival, but I want to win as much as possible. My mentality has always been as a winner, I have my dreams and then work to achieve them.
"My motivation is clear, I want to get back into Europe in one of the most important leagues in the world."