I grew up around agriculture, and have heard this as an expression. I think it had something to with spoiling the dairy cow’s milk, but that’s just a guess.
So, what’s so bad about cows gettin(g) in(to) the alfalfa?
Well in case you are interested, the alfalfa diet/milk connection has apparently been studied, with really really boring results: http://cat.inist.fr/?aModele=afficheN&cpsidt=18342372
Im guessing it has less to do with the taste of the milk and more to do with the purpose of the alfalfa. If a farmer is selling the alfalfa in bales it could have to do with not wanting the cows to obliterate a lucrative crop. I could be wrong.
Ruminants can get really screwed up by eating too much food that their digestive fauna go through easily, resulting in bloat. It kills the animal pretty frequently. I lost a market sheep this way, when I was a wee 4-Her.
ETA: Wiki sez:
Okay, you want me to order you a copy too? 
This.
FWIW, it’s not a pretty sight. Cow dead on its side, stomach split open, green stuff spilling out. It’s enough to make a 5 year old say “Eeeewwww”, or at least that’s what I said at the time.
Come to think of it, a friend had a cow get the bloat. The vet poked a big needle into her stomach to let the gas out. You could actually hear it. It was pretty worrisome, but the cow made it. This was one of those small farms where the cows were like pets. 
Horses too.
We ate the chickens. 
Our horse vet* claims that an ice pick to the stomach is a viable emergency solution. Ugh.
To quote my husband: “Animal husbandry is disgusting!”
*For horses I board, not that I own, so I’m not too concerned about his methods so long as I get invited to all the horse autopsies.
Which is why you should never let a cow eat a dog.
I grew up smack in the middle of about five states full of farmland and I never heard anyone say that.
Of course, I did spend a great deal of time as a child reading (for example, “Lord of the Rings”) 24/7. It could be I just wasn’t paying attention! 
Too late.
Oh, right, that’s only other cows.
I know that’s illegal now. Right?
That is what the expression is all about. Cows bloating. Dad use to use a sharp knift and stab the cow in the side to relieve the gas.
Any cows? 
Ever heard “tell him how the cow ate the cabbage” or something similar.
I’ve been a dedicated city boy for over forty years.
It’s kinda coming back to me, thanks to you guys, IIRC, cows love green alfalfa and won’t stop eating it.
On a milder note, I remember alfalfa causing the cows to have even worse-smelling flatulence than usual.
Pics or it didn’t happen.
On the damage-to-the-alfalfa side of things. I remember one time my grandfather let a neighbor pasture his cows in one of alfalfa fields for a brief bit. (Don’t know why the neighbor needed an extra field.) Afterwards, we went out thru the field to check for damage. Mainly looking for damaged furrows and cowpatties blocking irrigation. While I never heard the expression, I grew up knowing that cows in an alfalfa field should be avoided.
As for bloating, there’s a scene involving boated sheep in “Far from the Madding Crowd” where Alan Bates uses a tube/spike device to fix some sheep. The spike goes into the tube, both are shoved into the stomach, the spike is pulled out and gas escapes thru the tube. Disgusting … and fascinating.
Well, that settles that. The bear does not shit in the woods!