In "Hamlet" does Gertrude tell Claudius the truth about what happened between her and Hamlet? Is she following Hamlet's advice at the end of 3.4?

At the end of Act 3, Hamlet advises his mother to "by no means...let the bloat king tempt you again to bed" (III.iv.181-2) again, and to keep quiet about what they have spoken of that evening.  The queen promises that "I have no life to breathe what thou hast said to me," (III.iv.198) indicating that she will keep quiet.  And she does, for the most part, keep that promise.  She doesn't say anything about Hamlet's incensed ranting regarding her and her hasty marriage and hasty turning toward Claudius in her bed or in her heart.  We don't know if she keeps the first promise, about never going to his bed again, but we can only assume that she doesn't; she remains with the king, and seems to be happy with him throughout the rest of the play.  She does reveal that it was Hamlet who killed Polonius, but about the other things, she keeps quiet.  She mentions that Hamlet "weeps for what is done" to Polonius, and that he went to "draw apart the body he hath kill'd" (IV.i.24-27), which sends the King off on a hunt for the body of Polonius.

I hope that those thoughts help a bit; good luck!