2013 MacBook Air 13" won't boot - green MagSafe LED
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 - the only way to diagnose this is for the battery to run down completely and then connect to power with auto power on enabled and/or open the case and begin repair level triage / fault isolation / part replacement or repair ( hereafter called open and repair)
an issue with the SMC - the only way to diagnose this is to try one time to reset the SMC - if after one reset, it doesn’t power on, open and repair.
an issue with the logic board - open and repair
an issue with a failed battery dragging the power too low to start - open and repair


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.  This will triage and isolate the failure to the connectors running the keyboard and assure the logic board and storage and OS are all functional.

However, 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 at 2000 RPM on MacBook Pro and MacBook Air.

If you're not technically inclined, or it's the third option (likely), it's time to take it in for service to someone that has the tools and training to perform the open and repair steps.