Yes, you can. You have to use two expediting services to obtain both the FBI background result and the Apostille on the FBI result.
https://accuratebiometrics.com/wp-content/uploads/2024/12/Out-of-State_Forms-Packet_2024.pdf
You need to visit the District Police Station (Main ones). There is a department called 보안과 외사계 (Foreign Affairs Department), and they will observe and help you get your fingerprints completed and sign (stamp) on the paper to make the form official.
https://accuratebiometrics.com/services/international
NOTICE: Be aware that you can only download the result one time!!
https://monumentvisa.com/korvia-consulting-html/
You can submit electronic copies of an FBI background check for this service, then after they get an apostille stamp on the FBI background check result, they will deliver the result to Seoul (Korea)
Please read the article for more information
http://www.korvia.com/apostilled-fbi-background-check-korea/