Learn English – “Put the way forward “

OED mentions envy in passing under green and intriguingly it introduces yellow:

3. Of the complexion: having a pale, sickly, or bilious hue, indicative of fear, envy, ill humour, or sickness (also in green and wan, green and pale). Also in extended use. See also green sickness n.

• With reference to envy, chiefly in green with envy (also jealousy), green eye n. (yellow being the traditional colour of jealousy).

OED gives earliest citations, and Merchant of Venice and its "shuddering fear" appeared before Othello, but nothing before them:

2. fig. In and after Shakespeare.
a. Describing and alluding to jealousy. Cf. green adj. 3.

1600 Shakespeare Merchant of Venice III. ii. 110 Shyddring feare, and greene-eyed ielousie.

b. green-eyed monster n. (usu. with the) jealousy.

a1616 Shakespeare Othello (1622) III. iii. 170 O beware ielousie. It is the greene eyd monster.

Green with envy is a lot later:

1863 C. Reade Hard Cash xliii, The doctor was turning almost green with jealousy.

It's idiomatic, but it's not an idiom—a phrase or usage whose meaning is not determined by the meanings of its parts. You can use less than readily with just about any adjective which denotes a measurable quality:

less than eager
less than perfect
less than unanimous

As the Comments suggest, less than eager is ordinarily used in an ironically understated sense more or less equivalent to strongly averse.