The ques is that when we increase the speed of a ceiling fan do we increase the flow of electricity supply?

This sounds suspiciously like a homework question.

The answer is yes. The flow of electricity must be increased to have the fan motor increase the fan's speed.

Not exactly. For a PSC motor, one winding is phase shifted by a capacitor. The capacitor value is increased causing a larger phase shift. Smaller values of capacitance result in slower speeds.

The winding resistance for the winding excited by the capacitor is a little higher because the capacitance raises the voltage slightly for that winding.

This is only one way. Winding taps is another.

Need a high value capacitor