Yes you can! It takes a week or three to get really rotten and rich. I've done it for exactly the reasons you say: to dispose of noxious weeds and/or their seeds, and even just seedheads of plants I like but have too many of already.
I found a couple problems with it, but I think if you strain it as soon as it goes rotten and rich, the problems might be avoided.
=> It can really REALLY stink if left long
enough. Like Ugh!!! Wow. Impressive.
=> It attracts clouds of flies. I leave something over the top as a lid but it is never quite airtight or flyproof.
=> Last year I left it out of the way, fearing the stink, and then of
course I left it too long. When I opened it, it was swimming with some squiggy organisms that I can't identify and they really grossed me out. I dumped the whole
bucket and its contents in a dry ditch where the nutrients won't go to waste, but everything will dry up first.
So I think if you keep an eye on it and strain it promptly and use the juice, it's good stuff. Dilute it with other irrigation water, of course! If you're afraid the solids might still have unwanted seeds that survived the anaerobic soak, let them dry up for a few days and then add them back into the new batch.