"Quick" and "slow" always has to be compared to something. It is perhaps a shortcoming of most books on thermodynamics that they do not explicitly state what that something is, though perhaps it is also because it's a bit tricky to explain.
The scenario that one always imagines is a gas in a compartment in a piston. A realistic system of this sort will have an obvious time scale, on which heat will leave or enter due to imperfect insulation. For the adiabatic equation to apply, the change in volume will have to be done "quickly" relative to that speed.
There is however another time scale which occurs due to the gas having constituents and therefore having internal dynamics, such as eddies, or local density variations. Imagine pulling the piston out so fast that the wall moves significantly faster than the speed of sound in the gas --- you will create a vacuum which then causes a shockwave to be set up and some time will elapse before the system settles back into equilibrium. This will occur roughly on the order of size of box divided by speed of sound. For the "quasi-static" limit to apply, we have to move the piston slower than this speed.
For this system, hopefully you can see that if it is close to equilibrium, then the state of the system can be expressed as pressure and volume, and you get to ignore all the internal dynamics of the gas.
All applications of statistical mechanics and thermodynamics contain either explicitly or implicitly such a coarse graining of microscopic information, and therefore something of a floor on what the fastest macroscopic transition can be. A lot of confusion, e.g. over entropy, tends to occur when people forget this and start believing that coarse grained descriptions are somehow complete and exact.