Marquez, who won back-to-back Liga MX titles as club captain, was one of the standout performers of a Mexico side who somewhat surprisingly reached the last 16 of the World Cup and only narrowly lost to Netherlands.
"It is the player's decision to look for new challenges," a statement posted by club president Jesus Martinez on his Twitter account read. "We wish him the very best of success in his new club Hellas Verona in the Italian league."
The statement added that Marquez will always be welcome to return to Leon in the future.
Marquez also sent a message thanking the Guanajuato club's fans, claiming he would always carry them in his heart.
Marquez joined Monaco in Ligue 1 from Club Atlas in 1999, before signing for Barcelona in 2003 and going on to become one of the most successful exports in the history of Mexican soccer.
The centre-back left Barcelona following World Cup 2010 and endured a difficult spell at New York Red Bulls, before signing for Leon ahead of the Clausura 2013.
He lifted back-to-back Liga MX titles as captain of La Fiera, at a time when many thought his career was coming to an end.
Marquez's return to form opened the door to a comeback to the national team when El Tri was struggling in qualifying for Brazil 2014 and he duly delivered as captain of the side for a record fourth-consecutive World Cup finals.
Verona finished 11th in Serie A last season, just three points behind AC Milan and six behind Inter.
The club have also confirmed the signing of Antonio Luna from Aston Villa on a loan deal with the right to make the move permanent.