Kyrgios, 19, is the first Australian man for 10 years to reach the last eight and hopes to become the first for 39 years to win the title.
"Obviously the crowd will be right behind him, understandably so," Murray, 27, said.
"They're going to watch him play a lot of matches like this over the next 10, 15 years probably, and that's just something that I'll have to deal with in my way.
"I've played a lot of matches. I've played in the French Open against French players where the crowd can be very difficult. I've experienced it before, so hopefully I'll deal with it well."
The British number one will be playing in his sixth successive Melbourne quarter-final, while Kyrgios has matched his run at Wimbledon last summer, where he upset Rafael Nadal in the fourth round.
Dealing with an expectant home crowd is something Murray is well used to, and the Scot has been impressed with the way Kyrgios has used the support.
"When you have the crowd behind you it obviously helps, makes a difference," sixth seed Murray said.
"Especially if you're tired and a bit fatigued, the crowd can give you that extra lift and help as well. He's obviously handled everything very well so far."
The pair have met once on tour, with Murray winning in straight sets in Toronto last summer, and the Briton also won the set they played at the International Premier Tennis League (IPTL) exhibition in India two months ago.
"There were some things that I did in that match [in Toronto] which I believe will work on Tuesday," said Murray, who is on course to meet Rafael Nadal in the semi-finals and Novak Djokovic in the final.
"So I've learnt stuff from playing against him there and at the IPTL and watching him a bit, stuff that I feel I can use and exploit in his game."
Kyrgios pulled out of the Australian Open warm-up tournament in Sydney with a sore back and has continued to feel the issue throughout the tournament.
However, he was fit enough to come back from two sets down to beat Italy's Andreas Seppi in a thrilling match on Sunday evening, describing the experience as "crazy".
Asked to compare it to Wimbledon, he said: "I think this one, it feels a bit better, honestly.
"There was a lot of expectation coming into this tournament. I was obviously out for a couple weeks before Sydney. I wasn't expecting anything, especially not quarter-finals.
"It's just massive, especially to do it in front of your home crowd."
Kyrgios could not back up his win over Nadal at Wimbledon, going down in four sets to Milos Raonic in the quarter-finals, but he feels better prepared this time.
"I think I'll be pulling up better than I did at Wimbledon," the world number 53 said.
"I know what to expect now, what I am going to be feeling, especially after a five-set match like that.
"It's massive confidence being 19 knowing that you can last matches like that. It's massive."
Third seed Nadal or seventh seed Tomas Berdych will await the winner in the semi-final, with their match taking place earlier on Tuesday.
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