Green is traditionally the colour of jealousy as shown in the idiom green with envy and in this one - the green-eyed monster is jealousy personified.
This expression is a quotation from Shakespeare's Othello where lago warns : O! Beware my lord of jealousy. It is the green-eyed monster which doth mock The meat it feeds on.
RELATED IDIOMS :
green about the gills = green around the gills = green at the gills
looking or feeling ill or nauseous – informal
A person's gills are the fleshy parts between the jaw and the ears : this sense of the word dates from the early 17th century. Other colours are occasionally used to indicate a sickly appearance - much less common is rosy about the gills indicating good health.
permission to go ahead with a project
The green light referred to is the traffic signal indicating that traffic is free to move forward. Red and green lights were in use from the late 19th century in railway signals. But this figurative use of green light appears to date from the mid 20th century.
1997 - New Scientist - Zemin even got the green light to buy nuclear power plants.