I don't understand this part of the question.
but I can only connect it to the GPIO pins the LCD screen is already using.
Why only to those pins? Are the other GPIOs in use? Do you only have wires to those?
Without knowing the detail of your LCD connection I would say:
You can't arbitrarily multiplex LCD signals with a fan switch.
Knowing the protocol and all conditions under which you LCD is controlled, it might be possible
to use some pins combined with a decoder/statemachine to switch a fan on or off.
However from your remark "I'm ultra noob" I would guess that is beyond your capabilities.
A fan can be switched on/off from ANY GPIO pin, but it will require an extra driver like an NPN transistor or FET.
I am just about to write a multi page article on "how to control a relay from a computer" after seeing
that question for the hundredths time.