If your PS4 controller won't charge, you can try swapping out the charging cable or checking the charging port.
The PS4 is in the twilight of its life, gradually being replaced by the PS5 on a wider scale. However, the PS4's usefulness ...
To connect AirPods to PS4 or PS5, you'll need to pick up a Bluetooth adaptor with audio capabilities, since Bluetooth doesn't come with the consoles.