You can try a Power/Pressure Washer. I'm not sure if it's stained in the sense that there was a 'liquid' that got into the concrete. Yes, concrete is somewhat porous but perhaps it dried quickly enough that it's not that deep. A water hose only gives about 40lbs of press, give or take.
Have you tried scrubbing lightly (with a Nylon-bristle brush) with household dish detergent (like Dawn or Dove)? Other than that, you can try a 'true' concrete cleaner. They sell it at the Home Centers.
My best to ya and hope this helps.
Jay J -Moderator
PS: I'd start by pruning the trees, in the least. After that, if the nuts still pose a problem, you can prune some more or cut them down ...