Roy and Shirley Erskine, from Dunblane, said the letters had left them “upset”.
Their daughter Judy told them to bin the letters, which attacked the 29-year-old former Wimbledon champion’s behaviour on court and also criticised his accent.
Mrs Erskine, 84, told the Radio Times magazine: “We’ve had one or two nasty letters. One about Andy supposedly snarling on court, and another about his accent, saying he sounded American.
“Just comments like that, but they do upset you. Judy would say, ‘Put it in the bin, just forget it’. She would tell me that there are always going to be people like that.”
Her husband, 82, said he finds it hard to deal with “very nasty” online comments about his grandson and daughter.
He said: “The people who write these comments, that’s their prerogative, but I find it upsetting.”
Mrs Erskine said some of the letters provide light relief, with one man writing to tell them he thinks Murray needs to eat more baked beans.
Her husband said he is one of the harshest critics of Murray’s performance on court and prefers to watch him on television than live as he struggles to contain his language.
He said: “If I’m at home I can kick the coffee table and use some language that I probably shouldn’t.”
The couple revealed that while they struggle to go to the shops in their home town as so many people stop to talk to them about their famous grandsons, Murray - who sealed a record fifth title at Queen’s on Sunday - managed to take a run round the town virtually unnoticed on Boxing Day.
He ran from his mother’s home in Bridge of Allan to nearby Dunblane, disguised in a beanie hat, and then visited the tennis courts where he watched a family game before being recognised.
Mrs Erskine said: “Andy went on court and hit with them. That must have been a lovely moment for those kids.”