Can a quasar die?
Yes, a quasar will shut down when it runs out of material.
A quasar is the result of material falling in to the accretion disc of a supermassive black hole at the centre of a galaxy. The accretion disc is matter swirling into the black hokle. Material falling into the accretion disc gets heated by gravitational forces and friction until it emits a lot of energy.
When the material falling into the accretion disc runs out, the quasar will be stop emitting. It is possible for a quasar to reignite if more material becomes available.