To get a Green or Blue Chocobo, you need to mate Great or Good Chocobos to each other (so Great + Great, Good + Good, Great + Good or Good + Great). It doesn't matter what class they are but the better the class, the more chance of a 'successful' mating. A Class C chocobo will probably not get you a Green or Blue, but a Class B, A or S has a much higher chance.

To raise the Class of a Chocobo, you must go to the Chocobo Square in the Gold Saucer and talk to Ester about being a race jockey. Race your Chocobos through the lower grades until you're satisfied. It will only take 3 wins for the Chocobo to go up a Class so it's probably in your best interest to just keep going until the Chocobo is Class S.

There is also another vital thing to have in possession when planning on getting a Green or Blue Chocobo and that is the Carob Nut. You get these from the red dinosaur monsters near Bone Village. When you mate the Chocobos, Billy will ask you for a nut. Give him the Carob Nut because anything else will NOT give you a Green or Blue Chocobo.

For future reference. You mate your Green and Blue Chocobo together to get a Black (you must use a Carob Nut again). And to get the Gold Chocobo, you gotta mate the Black with a Wonderful Chocobo (which you can catch wild at the tracks near Icicle Inn) and for this mating you need to use a Zeio Nut (obtained from the Goblins on Goblin Island, a tiny island to the east of the farm).

Good luck.