If you mean the royals plotting against each other, that is most unlikely, for the following reasons:
In The Lion in Winter, the plot revolved around succession to the throne. The King, Queen, and their children were all plotting who would succeed the King. In TSL, the plot revolves around Graham, Alexander, and Rosella all trying to solve a problem for Valanice. So the plot does not lend itself to plotting, if you will pardon the pun.
It's conceivable that, in a subplot, Alexander and Rosella plot who will succeed to the throne of Daventry. However, as you get to play both Alexander and Rosella, this seems most unlikely, as you would get to play both sides of the conflict.
Since the only royal that you don't get to play is Valanice, then either the plotting would have to be against Valanice, or Valanice would have to be plotting against one or more of the other three. The former is unlikely, as the object is to help Valanice. The latter case, Valanice plotting against one or more of the other three, is the only possibility that seems plausible, yet I don't think that that is part of the story.
Of course, the evil characters will be plotting against the royals. And I think that it has already been stated that personal problems will be a big part of the story.