The 37-year-old pitcher was to travel from West Palm Beach, Florida, to New York for the procedure.
"Hopefully he can pitch again in about a week," Martinez said in a videoconference with reporters before Washington's exhibition game against the Miami Marlins. "We want him to get it taken care of now so it's not an issue."
Martinez said the Nationals "still have plans, as of right now, that he will start the season with us on his scheduled day, but we'll have to see -- after this procedure's done -- how he's feeling."
In 2006, Lester's rookie season with the Boston Red Sox ended early because he was diagnosed with a form of lymphoma.
He underwent chemotherapy and returned to the Red Sox at spring training before the next season.
Lester is entering his 16th year in the majors and his first with Washington. He was a free agent and joined the Nationals on a $5 million, one-year contract after playing the past six seasons with the Chicago Cubs.
The starting pitcher originally was scheduled to make his first appearance for Washington in a Grapefruit League game Thursday.
But Martinez said that, after additional test results came in, a decision was made Tuesday for Lester to have the surgery.
He said Lester had been feeling tired.
"That's the big issue. I feel like once they get this out, he'll have a lot more energy throughout the day," the manager said. "I hope it works out for him. I really do. He's a big part of what we do here, and we love having him. He's been working his tail off, day in and day out, and I know he's going to help us."