Green Singing Finches come from Africa where the seasons are reversed. Mine usually don't get interested in breeding until the end of summer / early Fall. In fact, mine hatched out young in December. Since many Green Singers were imported recently, yours may just not be in the mood to sing and breed right now - in their head, it might feel like the "wrong season". Be patient - give them the spring and summer off and they may surprise you in the fall. I think mine sing the most from mid-summer into the fall. Mine are
totally quiet now. Also, there's no need to separate them - the males sing just fine in the same cage with the female,
when it's the right time.
As someone else mentioned, your Green Singers may also start singing once they feel more relaxed. I have mine in a big cage in my sunroom where there isn't much human "traffic" - and they have the whole cage to themselves. Also their cage is up against a wall which makes them feel more secure. I now have even more plastic plants in the cage than in this photo (but they did breed with the cage like this). I also sometimes play a CD of jungle birds that makes all my finches active and chirpy. You might try that too.
I just re-read your post. Your cage sounds kind of small. They may never breed in a cage that small. You might want to consider moving up to a larger cage or aviary. Green Singers also do not mix well with other birds when breeding. They can either get aggressive with the other birds or simply not want to breed.
p.s. I assume you are aware that Green Singers use an open canary-type nest? They do not use nest boxes. And it helps to shield the nest area with plastic plants. They also like fresh egg food and live meal worms at breeding time. Giving them these foods in late summer may stimulate them to breed. Also, put lots of nesting material (jute and coconut fiber) on the bottom of the cage at breeding time. Having that seems to stimulate them.
Green Singers LOVE green food so make sure you provide some to yours a few times a week. I buy "mixed greens" type salad mix for mine and they eat it right up! You can also give them clean washed dandelion leaves, if you are sure they have no weed killer or animal urine on them.
