What does it mean to turn green with envy?

This phrase means to be jealous. The origin of the phrase is supposedly;

'Green' and 'pale' were different meanings of the same Greek Word. In the seventh century B.C., the poetess Sappho, used the Word 'green' to describe the complexion of a stricken lover. The Greeks believed that jealousy was accompanied by an overproduction of bile, giving a pallid green appearance to the victim.

Ovid, Chaucer, and Shakespeare did the same, freely using 'green' to denote jealousy or envy.

Maybe the most famous use is Iago's speech in Act 3 of Othello:

O! beware my lord, of Jealousy;

It is the green-ey'd monster which doth mock

The meat it feeds on.