Martin played with Tavares in New York for seven seasons before signing with Toronto on July 1, 2016, as a UFA.
"I don't want to say he's happy about John leaving, I'm sure he's not that type of guy," Martin said of Barzal. "But he does have a chip on his shoulder. I think he believes he can be one of the best players in the League. And you're going to have to have a bigger role to do something like that."
Trotz, who was hired by the Islanders on June 21, told SiriusXM NHL Network Radio on Thursday that he views Barzal as a "building block."
"I've talked to all the players about what I expect," Trotz said. "We're making some good headway. What's been a blessing, was when John made his decision, the players texted me saying we're going to be just fine. That's a huge step for any organization going forward.
"I think we're heading into the right direction. I know it's a big blow and everyone has the sky falling, but we're a young team. The future is really bright for the Islanders. We're right around the corner of doing something great."