Quadrature output refers to the phasing of the output signals. When the output signals, signal A and B, are 90 degrees out of phase with each other, the output is said to be in quadrature. This is the only thing that the term quadrature implies. (see What is a Quadrature output?)

x4 Logic denotes how the controller will interpret the signal that it is receiving. This is done by translating each edge of the pulse detected for the A and B channel into its own pulse. This translation takes place in the controller and not at the encoder.

This means if you order a quadrature encoder with 120 pulses per revolution, the output of signal A and B will be out of phase by 90 degrees. It does not mean that for every one revolution that the encoder makes you will get 480 pulses. The multiplication of the pulses only occurs at the controller