Gerrard, in an interview with Match of the Day magazine, said the new manager has already given Liverpool a lift after a disappointing seventh-place finish last season.
"It's a great club and after speaking with Roy Hodgson I'm really pleased to see his plans for the club. It's a new era now - we must forget about the horror of what happened last season and get this club back to where it belongs, which is the Champions League at the very least."
Gerrard said Cole, signed in July on a Bosman from Chelsea, has the ability to drive the team forward.
"[Lionel] Messi can do some amazing things, but anything he can do Joe can do as well, if not better. He used to shock us in training by doing footy tricks with a golf ball that most players can't even do with a football. I really fancy Joe for the [player of the year] award this season."