I recently noticed my driveway was wet even when it hadn't rained. It was even wetter the next day. I have realized there is a water flow occurring for about 1-3 seconds out of the house whenver the ground level toilet is flushed. I can't tell exactly where the water is coming from because it spouts out from a little rock wall that surrounds a garden. Any idea what this could be?

You have a pipe that appears to be loose or broken and the waste water is going out into the ground. If this is under the house, you may have it worst there.

Time to call a plumber. Unless you want to try and dig it out and find the leak