I had the same thing, if we jumped the quad or used a battery charger it would work. I orginally thought it was the starter solenoid of the key switch. However I eventually found a broken connection on the neutral (black) just behind the battery where it passes thru the plastic. I assume that the plastic either rubbed it, or was like that from the start, but years of vibration and water , finally killed the connection. Repaired and the bike is better starting now.
Start by cleaning all your connections. Starting from the battery terminals/connections then working your way to the solenoid/starter. As mentioned, also look for any loose or broken connections. If all that checks out, try pulling the starter and bench testing it. Another way you can start your atv without the electric starter is via the recoil starter (Pull Rope) if your atv is equipped with one.
First, check the battery. If it's fully charged, move on and check the carburetor. If it's not gummed up and seems to be in working order, check the ignition wire. If that's not the culprit, install a new spark plug and see if it'll fire up.