Murray disclosed that the Duke had asked him about his recent back surgery during their 50-second chat. The Scot also teased the Duke about his handwriting.
He said: "'He asked me about my back surgery which I had a few weeks ago and how I was feeling after Wimbledon, when there a lot of pressure on me.
"I teased him a bit as he had sent a note to me before Wimbledon wishing me luck and his wife had written me a letter afterwards. Her handwriting was beautiful and I told him how fantastic it was compared to his.
"I have met him a few times before and he has always been very easy going. He seemed very relaxed today."
Wimbledon champion Andy Murray, with his parents Judy and Will pose with his long time girlfriend Kim Sears (PA)
Murray's guests at the Palace were with his parents Judy and Will and girlfriend Kim Sears.
He said the Duke "seemed pretty relaxed" as he carried out his first investiture and "seemed to give everyone a lot of time".
He had been suprised he got to chat to the Duke for so long, as "I thought it would be about 15 seconds", but said their chat "flew by".
Murray, who said he does not expect to be back in full training for another four weeks, said he had no special plans to go out for lunch, as he was keen to get back into his rehabilitation training with an eye on taking part in the Australian Open in the New Year.
"To be honest we will probably just stop off somewhere on the way," he said. "I'm just keen to get back on the court."
Andy Murray's dog Maggie May wearing his OBE. (TWITTER)
Murray wore a Burberry suit for the ceremony, while Miss Sears wore a graphic print dress by Matthew Williamson and a beret by Jane Taylor.
After getting home he posted a picture on Twitter of his dog Maggie May wearing the OBE on her dog collar.
Murray was awarded the OBE in the New Year Honours, before his historic triumph at Wimbledon, after winning the US Open and Olympic gold medal in 2012.
The Duke was presenting awards to recipients from across the UK, including an OBE to Vicar Of Dibley producer Jon Plowman and an MBE to presenter Aled Jones.
The ceremony, which had been expected to last an hour, ran on to an hour and 25 minutes, as the Duke chatted to some recipients for more than a minute, far longer than the standard 30 seconds the Queen usually adheres to. As the ceremony went on, the Duke, perhaps conscious of the time, speeded up the process, with some recipients getting 25 to 30 seconds.
TV host Jones was recognised with an MBE for services to music, broadcasting and charity.
The former teenage choirboy, who now fronts morning show Daybreak, found fame at the age of 12 with his top five hit Walking In The Air, which was written for the animated movie The Snowman.
Plowman served as the BBC's head of comedy for 10 years, producing classics such as Absolutely Fabulous, French And Saunders, Vicar Of Dibley, Little Britain and The Office.
He was also an executive producer of the Olympic satire Twenty Twelve, which won a Bafta award.
Andy Murray's terrific win at Wimbledon was sealed with a balcony kiss with his girlfriend Kim Sears (Getty/Rex)
The Queen hosts the large majority of investitures but shares the role with the Prince of Wales and the Princess Royal.
Around 25 investitures are held each year, with most staged in the ballroom at Buckingham Palace, but others take place at the Palace of Holyroodhouse in Edinburgh and at Windsor Castle.
Before the ceremony the Duke practised ''dubbing'' a servant with a sword, ready for the process of knighting people. During the investiture he used a sword used by George VI when he was Colonel of the Scots Guards.
Sir Kenneth Gibson was the first person to be knighted by the Duke. Sir Kenneth, who has worked in education for 35 years and is currently executive head of several "challenging" schools in the north east, described the Duke as a "complete natural".
"He seemed very relaxed, far more so than myself," he said. "I found it all very overwhelming and was very nervous but he took it in his stride.
"I told him that it was a huge privilege to be the first person to be knighted by him and he acknowledged it with a smile.
"I know there was some speculation in the press about him practising so that he didn't chop anyone's ear off but he was absolutely fine."