I've got a green fig tree too (as of 9 months or so) and was unsure how to pick a ripe one too, but I've had plenty of practice now. Just wait until they give slightly under pressure from a gentle squeeze, similar in texture to brown figs but usually without any signs of browning.
In my case the green figs were ripe at the same size range as with brown figs.
If you do see one that's got some hint of brown it's probably overripe and may be starting to rot on the tree.
After picking the first one and testing the sweetness, you may prefer to let them hang until they're a bit more "squishy to the touch", but that's up to your taste.