Toilet Supply Valve Not Turning Off
The proper way to address this is to turn off the water and replace both valves.

EVERY HOME OWNER should know where the main water supply enters the house and where the valve is located in their home. It may be coming out of the ground in a crawl space, in a wall behind an access panel, in the utility room where the water heater is or elsewhere.  


  If there is very little water coming out from the supply valve could I
  still install/hook up the Fill Valve?


Yes IF the flow is slow enough that you can capture it in container before the container overflows. 

If the flow is significant then it is not practical. 

I have done this more than once. Leave the supply hose attached to the supply valve and put the end that connects to the fill valve in a bowl or bucket. This may be difficult in the small space between the toilet and the wall. You may need a small bowl and that could result in having to empty it frequently while you are changing the fill valve. Fortunately you have a many places to empty it in a bathroom. 

Doable but not exactly fun, two containers so you can have one catching the flow while you empty the other. 

Install a new 1/4 turn supply valve and a new braided supply line if it is as old as the valve. 

Edit: Jack makes a good point about the age of the valve. If your main valve is a gate valve it is probable that it may have issues shutting off completely or getting it open again. The only way to change that valve is to call your water utility company and have them shut off the water supply at the curb main shut off. Then you can install a new 1/4 turn valve, much nicer. If you do get it back on make sure you open it all the way, counter clockwise until it stops. It can leak if you do  not. I flooded a basement once because i did not open it completely and it leaked, opening it completely stopped the leak.