Let me guess, the 6-year-old boy is the firstborn?
Seriously, if he is, I have never figured out why the firstborn is so important, so precious, that all the other grandchildren are relegated to the background (by some grandparents). Seriously, did the grandparent do that to their own children, pick a favorite and then ignore all the other kids? Is the child's mother or father the favorite of the grandparent? Many times, if the grandparent has a "favorite" child of their own, then when that child has their own children, they sometimes will pick out a "favorite" there as well. Which I've never understood, because all children should be equal.
This is actually quite harmful to that young girl's psyche. If you have a good rapport at all with either the mom or the dad, I would suggest pointing out what it is you see during the day. (and perhaps it would be best if you spoke to the parent whose mother this is not.) Perhaps they don't know it's that bad. Or they have blinders on and refuse to see it, but someone bringing it to their attention might help them open their eyes and help their daughter. The parents need to know what it is that "grandma" is doing while she's home alone with the kids (and you're there), how she treats the kids, what she does that isn't equal between them, etc.
Seriously, this problem needs to be nipped now, not later. The girl is already showing obvious signs of starting to hate her grandma. I can't imagine ANY grandmother would want any of her grandchildren to not like her. Then again, she probably has no idea of how her actions are affecting this child. IF she does know and is deliberately doing it--then she needs to stay away from the children. And the parents need to man up/grow up and stand up to the parent (grandma) and say enough is enough!