Citizens and foreign citizens who have been granted temporary residence or permanent residence who have been fully vaccinated in the Republic of Serbia have a certificate issued by a health institution or body.
Crew members and travelers transiting for less than 12 hours, children under age 12 accompanied by parents who have valid PCR test results and properly accredited diplomats, international organizations and military undertaking official travel and their families are exempt from presenting a COVID-19 PCR test result.
Travelers entering from North Macedonia, Bosnia and Herzegovina, Montenegro, Albania, or Bulgaria are also exempt from presenting a COVID-19 PCR test result.
Children under the age of 2 are exempt from presenting a COVID-19 test.
Travelers with proof of recovery from the virus within the last 90 days prior to boarding. Proof of a recent positive viral test and a letter from your healthcare provider or a public health official stating that you were cleared to travel.
More info at the CDC site
Travelers who did not get tested, are advised to stay home and self-quarantine for 10 days after travel.