No it doesn't. The radiation ionizes the gas inside allowing electricity to conduct from the from the cathode to the anode. Note in your picture the 500 volt power source. The radiation only usually ionizes one atom, but the applied voltage differential causes a
electron avalanche which allows the gap to be bridged which sets off the counter. The electron avalanche is caused by the radiation, but it is not powered by it, the battery does that.
If you want to capture energy from radiation, you simply place enough matter in front of it so it will interact with it, converting it to heat. What you do with that heat is up to you.