I live on a dairy. Yes, when our cows grazed on our cover crop which included radishes, it affected the flavor of the milk. And the color, too.

We actually had some customers tell us later that they wouldn't have kept buying our milk due to the unpleasant flavor, except that they knew it was temporary.

I'll second this; my parents had milk goats while I was young, and their diet definitely affected the milk flavor. Garlic, onions, even some weeds would do it.