If you have landed at Incheon, your choices for a shower are limited.
Spa on Air is one choice located on B1 of the arrival hall. It is open 24 hours and there is a small fee to use it. You might also consider using the
Capsule hotel located near the airport train area. This also has a fee. These two facilities are located outside of baggage claim.
If you are leaving Incheon airport (flying to another destination) and have gone through security, then there are shower facilities inside, although they may charge a small fee for the towel. There are also lounges that may have showers. These facilities are not available for arriving passengers. If you are transiting, then you can access it. But if you have flown into Incheon airport and are not transferring, there will be no way for you to get to these showers.