Quarantine is not advised for vaccinated individuals. There is no clinical evidence suggesting such people can transmit the virus unless infected.
After vaccination, if someone gets infected, symptoms will be milder and are called a breakthrough infection. Studies show that first-dose vaccinated people who test positive had lower virus levels in their bodies than unvaccinated positive people. Hence, researchers state this decreased viral load among vaccinated people might be less infectious as they have fewer viruses to spread to others.
However, there is no evidence to show that a vaccinated individual who is neither infected nor symptomatic would transmit the virus to others. Furthermore, such vaccinated individuals need not quarantine themselves as no medical bodies have advised doing so. But, vaccinated individuals must follow the same preventative measures as others until they are deemed "fully vaccinated." These precautionary measures are advised not because vaccinated people can transmit the virus but to ensure that they don't contract the viral infection from others.