New Jersey doesn't have any special criteria outside of recommendations for domestic travelers
Unvaccinated individuals who do travel should get tested 1-3 days before their trip and 3-5 days after travel. Even if they test negative, the CDC recommends unvaccinated individuals stay home and self-quarantine for 7 days after domestic travel, and for 10 days if they are unable to test.
Fully vaccinated individuals and those who have recovered from COVID-19 in the past 3 months do not need to get tested or self-quarantine after travel. However, the CDC recommends fully vaccinated individuals self-monitor for COVID-19 symptoms, and isolate and get tested if they develop symptoms.
Source : NJ COVID Site
But, some states may have different rules in place