In Shakespeare's Hamlet, the relationship between Claudius and Gertrude is as strong as ever in Act 4.1--at least on the surface.
She immediately tells Claudius exactly what he asks for: what do her "sighs" and "heaves" mean and where and how is Hamlet (Act 4.1.1-5)?
She tells him that Hamlet is as
Mad as the sea and wind when both contend
Which is the mightier. (Act 4.1.6-7)
And she informs Claudius of the details concerning Hamlet's killing of Polonius, and that Hamlet is removing the corpse from her chamber.
But she doesn't stop there. She also tells Claudius that Hamlet "...weeps for what is done" (Act 4.1.27)--is sorry for killing Polonius.
Like much in the play, however, Gertrude's actions are ambiguous, and thus, while the relationship between her and Claudius appears to be as strong as ever, there is a possibility that it is not.
Gertrude, in fact, tells Claudius exactly what Hamlet asks her to tell him just a few minutes before Act 4.1:
[Don't let Claudius] Make you to ravel all this matter out,
That I essentially am not in madness,
But mad in craft. (Act 3.4.190-193)
Hamlet tells her to not let Claudius sweet talk her into telling the truth about Hamlet's madness--that he is only acting. And she doesn't. She tells Claudius that Hamlet is as mad as the wind and the sea when they contend to see which is the most powerful. Then, as mentioned, she also defends her son and tries to alleviate the seriousness of what he's done by telling Claudius that Hamlet is sorry for killing Polonius.
Her words are ambiguous, because we don't know her motivation. She may tell Claudius that Hamlet is mad because Hamlet asks her to, or she may tell Claudius that Hamlet is mad because she really thinks he is.
Gertrude promises Hamlet that she will do as he asks:
Be thou assured, if words be made of breath
And breath of life, I have no life to breathe
What thou hast said to me. (Act 3.4.200-202)
She will not speak the truth about Hamlet's madness. And, again, she doesn't.
Yet, after seeing Hamlet kill Polonius, then speak to the air as if there is someone present (he sees the Ghost, she doesn't), the possibility certainly exists that she is merely saying what Hamlet wants her to say, and that she still believes Hamlet is, indeed, mad. And one could hardly blame her.
Thus, the relationship between Gertrude and Claudius might be as strong as ever in Act 4.1, or a rift may have developed: Gertrude may have switched her allegiance from Claudius to her son, Hamlet.