Can international students in South Korea apply for permanent residence if they stay in South Korea for 5 years? Or do you need to find a job after graduation to qualify?

The only visa that Korea can apply for is a sovereign visa, D-2 to F4 - after 5 years, it will be converted to F-5 (so-called permanent residence)

D-4 is in South Korea Language student visa for language reading. D-2 refers to the study abroad visa, which includes undergraduate, master's and doctorate.

F4 refers to domestic Koreans who have graduated from a 4-year or above formal undergraduate program in China, or who have graduated from South Korea with a bachelor's degree or above.

For domestic ethnic groups other than Koreans, the only way to stay permanently in South Korea is to get married.