Royle, who was the last coach to win a trophy for the Toffees in 1995, believes that the Spaniard will be confident of his chances after leading Wigan Athletic to FA Cup glory last season.
"Roberto comes into the job with a trophy under his arm, which is nice," Royle told Everton TV. "It's an exciting time.
"I'm constantly referred to as Everton's last manager to win a trophy and I'm fed up with it. Please Roberto, take it away from me, I'd love you to win a trophy."
Martinez has replaced David Moyes, who failed to pick up silverware during an 11-year stint at the club.