You can't, we only know the region.

Inspired by strolls round Greyfriars’ cemetery JK Rowling looked through the register of interments and found Wormtail's surname. It was a perfect resource.

Greyfriars Kirkyard facing North East towards Candlemaker Row and Peter Pettigrew

Greyfriars cemetery dates back to Mary Queen of Scots, she gifted the land in 1562 because the existing cemetery beside St Giles Cathedral on the busy Royal Mile stank to high heaven all summer.

There are three entries for Pettigrew in the Register of interments in the Greyfriars burying ground, Edinburgh, 1658-1700 The last is the most tantalising

Malcolm, weaver, his wife, Agnes Pettigrew
John, elder, inn dweller, his wife, Margaret Pettigrew 10 Feb. 1673
John, inn dweller 28 May 1674
See also Peter.
Pettigrew (Petecrew, Pitero, Pitekreu, Pitticrew, Piticrow,
Pitiecrue, Pitiekren), Agnes, wife of Malcolm Alexander, weaver 20 Aug. 1662

“See also Peter”! The registry is dated before Scotland and England united, so a good number of the names in the Wizarding world have a pre-Enlightenment Scottish flavour. Given Rowling’s desire to make the wizarding world feel ancient the register was like a gift from Dumbledore.

The theme of Peter Pettigrew’s plotline is loyalty and duty.

The Marauders, Hogwarts bffs James Potter, Sirus Black, Remus Lupin and Peter Pettigrew with the Marauders map in the background
The Marauders: Peter Pettigrew, James Potter, Sirus Black, Remus Lupin

Peter betrays his friends James Potter and Sirius Black and Dumbledore's Order of the Phoenix for the protection of the powerful Lord Voldemort.

In The Goblet of Fire Pettigrew loyally sacrifices his own hand to resurrect his master Voldemort. Voldemort provides a replacement silver hand, he needs a fully functional servant.

During the Deathly Hallows, Peter hesitates to kill Harry, out of mercy, his hand turns loyal to Voldemort and strangles him. Decency punished, true horror. In the films house elf Dobby merely knocks Peter unconscious, being strangled by your bewitched metal hand isn’t PG13.

Hunting Peter Pettigrew’s grave in Greyfriars Kirkyard

Over a quarter of a million people have been buried in the Kirkyard, but it’s only 5 acres so not all of them had tombstones and many tombstones have been supplanted; so we don’t know precisely where the Pettigrews are buried.

Map of Greyfriars Kirkyard bordering Candlemaker Row  

We do know that it’s probably in the North East region bordering Candlemaker Row, because that’s where people were buried in the 17th century. It's perhaps appropriate that unloved Peter has an unmarked grave.

  is this Peter Pettigrew's rain eroded grave in Greyfriars Kirkyard?
Has rain erased the Pettigrew family's grave?

If you walk down Candlemaker Row there are a string of shops some with Harry Potter merchandise, and it once staked a claim as being Edinburgh's second "Diagon Alley".

But amidst the gloom of winter I had a little boy tell me Candlemaker row reminded him of dark wizard’s Knockturn Alley, with it’s crowded buildings, black doorways, and basement stairs twisting into gloom. So you could say that Peter Pettigrew was chucked in the ground behind Knockturn alley.

Clubbers returning home on dark winter nights report hearing the tinkle of metal fingers scrabbling across stone, but that's probably just tinnitus. Come visit Edina, but watch out for a lively silver hand.

The Marauders with map illustration by Adam Withers & friend Register of interments in Greyfriars Kirkyard