Jealousy and envy are technically different but normally used interchangeably.
The phrase was used by, and possibly coined by, Shakespeare to denote jealousy, in The Merchant of Venice, 1596:
Portia:
How all the other passions fleet to air,
As doubtful thoughts, and rash-embraced despair,
And shuddering fear, and green-eyed jealousy! O love,
Be moderate; allay thy ecstasy,
In measure rein thy joy; scant this excess.
I feel too much thy blessing: make it less,
For fear I surfeit.
In Othello, 1604, Shakespeare also alludes to cats as green-eyed monsters in the way that they play with mice before killing them.
Iago:
O, beware, my lord, of jealousy;
It is the green-eyed monster which doth mock
The meat it feeds on; that cuckold lives in bliss
Who, certain of his fate, loves not his wronger;
But, O, what damned minutes tells he o'er
Who dotes, yet doubts, suspects, yet strongly loves!