Is there any evidence that the Milky Way could have been a quasar in it's early history? Is it thought that most galaxies come from quasars?
A quasar is simply an active galactic nucleus (AGN) that is viewed from a particular angle; see the picture below, in which quasars are labeled "QSO". This is really a remarkable figure because historically all of the names in the figure were thought to correspond to different types of objects, when really they all refer to the same thing!
Your question really shouldn't be "Was there ever a quasar in the Milky Way?", since the dotted line in the figure would correspond to the Galactic plane and we would not see Sagittarius A* (the Milky Way's super-massive black hole) from the correct angle. A better question might be, "Has Sagittarius (Sgr) A* ever been active?" The answer to that question is yes; according to this page it was probably active (very bright with a jet) about 10,000 years ago. However, at the moment, it isn't really doing anything, since it isn't currently accreting anything (to put it plainly, it isn't eating anything, so it doesn't have enough energy to be active). However, many astronomers (myself included!) are anxiously waiting for a cloud of gas called G2 to fall into Sgr A*. We are hoping that Sgr A* will burp or do something interesting.
Such a dense region or black hole is thought to form by accretion of dust, gas and stars. This accretion process of a supermassive black hole releases huge amounts of energy emitted perpendicular to the accretion disk. If such a jet at the center of a distant galaxy points towards Earth we may see it as quasar.
Hence the answer is most likely yes, the Milky Way or some of its predecessor galaxies will probably have had quasars at their centers, at some period when consuming lot of material, and seen from appropriate direction.
Most galaxies are thought to contain a supermassive black hole at their center. Therefore the same applies to most galaxies.
Here two nice simulations: