Silverstone is the scene of Hamilton and Verstappen's dramatic collision at last year's race, an incident which heightened tensions between both camps during their title fight.
Verstappen went on to controversially beat Hamilton to the championship on the final lap of the final race in Abu Dhabi.
The Dutch driver was booed by fans in the grandstand opposite the start-finish straight when he spoke to F1 TV shortly after qualifying in second position.
Hamilton, who received a similar reception at several events last year, said he did not agree with it.
"I think we're better than that. I would say we don't need to do the booing," Hamilton said.
"We have such great fans, our sporting fans feel emotions up and down, but I definitely don't agree with booing. I don't think we need to do that.
"I think we should be here pushing everybody. It doesn't make any difference when you boo someone, they've already made the mistake, or whatever it is.
"But I really do appreciate the support I have here. Maybe some of them are still feeling the pain from last year, still."
Last year Verstappen said there was no point telling Dutch Grand Prix fans to not boo Hamilton as it is a natural part of sporting occasions.
Verstappen brushed off the negative reaction, saying his only frustration was being unable to hear Billy Monger's questions.
"It was a bit disappointing because I couldn't really understand Billy," he said. "If they want to boo me, they can do that. For me it is not going to change anything.
"I'm always happy to be here, it's a great track, it's a great atmosphere in general. Maybe some of them don't like me, that's fine, they all have their own opinions. I don't care."
Mercedes boss Toto Wolff shared Hamilton's view, saying: "I think we shouldn't be seeing any booing in any sports. I think that's unsportsmanlike.
"It's clear we love the support the drivers have here and the enthusiasm, that's fantastic. But if you are not into the other guys I think it's best to remain silent. I think that's the best way. The booing, I think none of the drivers deserve booing, no matter what happened last year."
A heavy downpour during qualifying had propelled Hamilton, one of F1's best drivers in the wet, into contention for pole position.
The Mercedes driver found himself in second position on two occasions as the conditions kept changing, although he could only manage fifth position when the session finished.
Hamilton said he and the team misjudged the timing of a battery recharge and failed to take advantage of optimum track conditions.
"I was really, really so hopeful because we have this incredible crowd and all of a sudden I was in the fight. It was feeling great. I was like 'I can definitely go quicker than that', I think I was two tenths up and then they said back off and go again. The next lap wasn't as good."
"We wanted to go to a more racey power mode for the final lap and charge the battery. As we did that it rained a little bit more on the final lap."