Inform on someone to the police.
In 2005, British newspapers picked up on a story about a burglar who had stolen cash, jewellery and an African Grey parrot from a house near Hungerford, Berkshire. David Carlile, widely described in the press as 'feather-brained', explained to the police that he knew that African Greys could talk and he didn't want the bird to 'grass him up'. Presumably, had the parrot been a Norwegian Blue, he would have left it to pine for the fiords.
'Grassing up' has been a commonly used expression in the UK since the mid 20th century, but is less common elsewhere. The first known use of 'grass' in that context is Arthur Gardner's Tinker's Kitchen, 1932, which defined a grass as "an informer". Grass was a well-enough established word in the 1980s to have spawned 'supergrass', i.e. a republican sympathiser who later 'turned Queen's evidence' and informed on the IRA, and which gave the Brit-pop band Supergrass their name in the 1990s.
Informers are variously known as squealers, noses, moles, snouts and stool pigeons. These terms invoke imagery of covert snooping around and of talking. Grass is less intuitive. It could just have arisen from 'snake in the grass', which derives from the writings of Virgil (in Latin, as 'latet anguis in herba') and has been known in English, meaning traitor, since the late 17th century.
There is another route to the word and this is via rhyming slang. Farmer and Henley's 1893 Dictionary of Slang defines 'grasshopper' as 'copper', i.e. policeman. The theory is that a 'grass' is someone who works for the police and so has become a surrogate 'copper'. The rhyming slang link was certainly believed in 1950 by the lexicographer Paul Tempest, when he wrote Lag's lexicon: a comprehensive dictionary and encyclopaedia of the English prison to-day:
"Grasser. One who gives information. A 'squealer’ or ‘squeaker'. The origin derives from rhyming slang: grasshopper - copper; a 'grass' or 'grasser' tells the 'copper' or policeman."
That comes only a few years after the term grass was coined and there seems little reason to doubt it as the derivation. The original users of the term 'grass up' were from the London underworld and would have certainly been better acquainted with rhyming slang than the works of Virgil.
Some have also theorised that the term 'shop', meaning 'give information that leads to an arrest', derives from the same source, i.e. that, as 'grass' derives from 'grasshopper', then so does 'shopper'. The earliest known use of shop in that context dates from around the same time as the emergence of grasshopper. The issue of the magazine Tit-Bits for May, 1899 includes:
"[He] volunteered for a fiver to 'shop' his pals."
As far as we know, African Greys don't go shopping.