Punctuation questions again. I feel like when I hear/see this performed it is almost always spoken with a period. I’d need to do a more thorough edition comparison – but it may well be an editorial choice to make this a real question.
The difference in meaning would be significant. One assumes that Gertrude does, in fact, know about the trip to England – the other leaves it more open. I don’t think Gertrude probably knows about the underlying purpose of the trip to England but it is clearly a bit of a sore point between everyone.
Hamlet may have a sense of the intended reason for the trip but this question could be a probing of the levels of awareness Gertrude has.
“You know that.” Assumes knowledge.
“You know that?” Assumes nothing.
The Hamlet Project 