The problem is unrelated to the status of your MagSafe light. Green means "charged" and amber means "charging." Seeing a green light on your MagSafe means it's getting the signal that they battery is fully charged; there's no other diagnostic info related to this light.
The problem you are having is that your Mac won't turn on. This could be due to:
- an issue with the power button on the top case assembly
- an issue with the SMC
- an issue with the logic board
If you have some technical skills, you can DIY diagnose the first two by bypassing the button and shorting the power pads on the logic board to simulate a button press or disconnect the battery for 30 seconds to reset the SMC.
The key to this that points in the direction of a logic board failure is this statement:
There's no fan noise...
Regardless of any other issues, if the logic board is functioning properly, the fan will spin.
If you're not technically inclined, or it's the third option (likely), it's time to take it in for service.