What is a galaxy that was once a quasar is likely have?
A galaxy which was once a quasar is likely to have a supermassive black hole at its centre.
A quasar is a young galaxy which has a supermassive black hole at its centre along with a lot of gas and dust. The material falls into the black hole creating an accretion disc. Material falling into the accretion disc gets superheated by friction and gravity to the point where it emits vast amounts of radiation. This is a quasar.
When the material around the supermassive black hole runs out the galaxy stops being a quasar. It will retain the supermassive black hole unless it is removed by a collision with another galaxy.