Actually luckypearl is closer to the answer by mentioning power efficiency.
The classification of amplifier is based on power efficiency and not Q-point. The Q-point is merely an indicator by means of DC conditions to determine the choice of amplification that gives a good tradeoff of signal amplification and power efficiency.
Q-point = Quiescient Point, the point when the system is quiet. In other words, the system is still consuming power when it is not amplifying any signal at this time. The higher the Q-point, the greater the power consumption when it is not amplifying any input signal.
Class A is used to describe an amplifier that amplifies the input signal without any distortion by using a full 360° cycle of amplification. The cost of such full-cycle amplification consumes higher power, hence lower efficiency. This is the reason the Q-point of Class A amplifier is usually a mid-point along the DC loadline, hence resulting higher ICQ, thus higher DC quiescient current, thus lower power efficiency. Class A usually has at most 20% or lower power efficiency.
Class B uses Push-pull 180° cycle of amplification. Thus results lower power consumption, hence higher efficiency of usually 50%. However, the Push-pull switching effect causes cross-over distortion. Push effect from amplifier to load, and Pull effect from load to amplifier.
Class AB is an improved version of Class B by using diodes to compensate the Vbe voltage drop at the output stage overcome cross-over distortion.