Do quasars exist today? Are there likely to be any quasars right now, or are all the ones we're seeing old galaxies today, like the Milky Way?
 A: There is no doubt at all that quasars are much rarer in the present epoch than they were at earlier epochs.
The requirements for a quasar are that (a) you assemble a supermassive black hole and then (b) that you feed it with material (gas), such that a large amount of accretion energy is radiated.
Quasar activity peaked at moderate redshifts ($z \simeq 2.5$) due to the required feeding processes of active galactic nuclei and the competition between the merger activity of gas-rich galaxies and the quenching caused by massive star formation and negative feedback from the AGN themselves. It appears that the "sweet spot" for comparatively short-lived phases of "quasar activity" is at redshifts of 2-3 where there was significant merger activity and the transport of gas into the central regions of galaxies, but that there had been insufficient time to fully exhaust the gas in galaxies with central black holes.
In the present epoch, many galaxies have exhausted their gas, and the expansion of the universe means that mergers are infrequent. That is not to say that quasars do not exist in the present epoch, just that their spatial density is so low that we do not have one in our close vicinity.
A: There certainly are quasars. Obviously, we can't see any as they are today because they aren't nearby. Quasars are a type of Active Galactic Nucleus(AGN), which means we won't find any within our local few million lightyears. However, as the Wikipedia page on Quasars will tell you, any time a supermassive black hole of a galactic nucleus gets a massive infusion of new matter, there's a good chance for a quasar to form. When I say massive infusion of new matter, I mean something on the scale of a galactic merger. In fact, many astrophysicists believe the Milky Way-Andromeda merger in 3-5 billion years will create a quasar.
Furthermore, the closest quasar we observe is only 600 million light years away. That means, because of expansion, the light we see from it is less than 600 million years old. To put that in perspective, trilobites first made an appearance about 520 million years ago, so this quasar was emitting light at about the same time that early arthropods were first walking on the seabed. That may seem like a long time ago to most people, but the universe has been around for billions of years. 600 million is not that long ago.
You also have to keep in mind that there are hundreds of billions of galaxies, which means substantially large chances all the time that galaxies will merge and create a quasar. We won't see most of them because they need to be pointed almost right at us. But most assuredly, there are quasars around today. With new ones being made and with ones we can see today being so recent, we can be sure they're around beyond a reasonable doubt. And let's take as a blessing the fact that the ones we do see are so far away. You wouldn't want a quasar nearby and pointed right at you. That's a recipe for bad stuff to happen.
