[rufus_t]

No. Like in English, it's "Gräset är alltid grönare på andra sidan"/"the grass is always greener on the other side". You should interpret it as "the grass on the other side always looks greener than it actually is". Your version is sometimes used as an after-the-fact statement that something didn't work out as planned, but it's not the traditional form.

[ck2]

According to wikipedia, French has this expression too.

Fascinating. Must be a very old expression.

[pbhjpbhj]

I thought it was from the story of the "Three Billy Goats Gruff". I don't know if that is the origin of if the story was built around the idiom. Seems a possible etymology though.