First thing to do is (or attempt to) charge the battery and when fully charged, tested.
If the battery is dead, trying to charge by running the vehicle at min 2500 RPM will take 24 to 72 hours or more depending on the state of discharge of the battery.
Jumping to start is fine, but if it will not run with the jumper removed, the battery needs to be charged or replaced before attempting to run the vehicle. The 20 minutes you let it run probably resulted in 90% of the charge rate going to charge the connected battery and not the battery in the quad.
When you disconnected the jump battery and tried to throttle up - if the battery in the vehicle was not charging (accepting charge) all the output of the alternator was being absorbed by other electrical components - when the battery is very low, it results in high current in resistive loads (lights, hand warmer, winch etc.) - if you install a 6 volt battery and run the engine, you will burn out the resistive loads before you damage the battery.
If you did not damage the CDI, charging system or lights, consider yourself lucky. You will probably need to get a new battery and determine why the battery went dead. You might have a shorted out rectifier, left an accessory on, have a defective key switch or a wiring problem. Whatever caused the battery to go dead (unless the battery simply failed) needs to be corrected or a new battery will do the same thing.
Shop Owner and Mechanic with over 50 years experience