I was wondering the same thing as LdiJ is asking. I think its absolutely possible that an adult would try to manipulate a child. I grew up in a pretty F-ed up family, so I know all about manipulation and lies.
If this is your mother-in-law, then she may resent you because you're not her ideal daughter-in-law (or for a billion other reasons!).
If this is your mother, she may be thinking payback...or is pissed that you didn't take her advice on something, who knows. People do the strangest things for the strangest reasons.
Its also possible that Gma said something about your daughter being heavy and then some snotty comment about how her cooking is better than yours and your daughter got the completely wrong idea of how to handle the situation. When a child is so young, they're minds are not capable of understanding the complexities of a simple statement. And they're so impressionable! If I tell my kids that I like a movie, they pipe back "That's my favorite, too!" Haha.
Have you talked to Gma? Have you asked her what her side of the story is? I would stop visitations and start counseling immediately. But I would also give Gma a chance to stand up for herself, unless of couse, she has a history of this type of behavior. If that's the case, then just don't let her see your daughter. Gparents, unfortunately and fortunately, don't have too many rights. She could try to get court-ordered visitations, but they would most likely not be granted.