Basic circuit flow of electrons
"how come the electrons quantity increase in a closed circuit conductor"

It is not the quantity of electrons in the conductor that changes, in fact that changes very little. As you said, it is (my emphasis)

"quantity of charge passing through a cross sectional area per unit (of) time"

In the water equivalent (which sort of works, but don't let yourself be carried away by it; it is just an analogy, not a reliable model) consider a water pump, with a hose connecting the output back to the input. Pump and hose are filled with water. The amount of water never changes, but when you crank up the pump, the water will circulate faster: more water passes through each cross-section of the hose per unit of time.