Did Queen Gertrude know about the murder of Old Hamlet?If she did then did she know who was the murderer?

It's impossible precisely to say. There is a case to be made in the text for either possibility.

To argue that she did know about Old Hamlet's murder, people turn to the evidence that her marriage to Claudius, she says, is "o'erhasty" - as if they should have waited to allow suspicion to subside? You can also play her reactions in the play scene as deeply uncomfortable ("the lady doth protest too much").

To argue that she didn't know, productions (such as the recent staging by Sir Trevor Nunn at London's Old Vic Theatre) simply have to play Gertrude as not very bright, and not very knowing. So Gertrude is horrified at the play, but doesn't put two and two together, and - potentially - only realises during the closet scene (Act 3, Scene 4), which then (in some productions) leads to her deliberately drinking the poison in the final scene.

Even the ghost is ambiguous - is she to go to heaven, or go the thorny way:

Taint not thy mind, nor let thy soul contrive
Against thy mother aught. Leave her to heaven,
And to those thorns that in her bosom lodge
To prick and sting her.

You can do it both ways. If you wanted a definite answer, it depends on how you read the closet scene, and Gertrude's behaviour after it. When Hamlet calls Claudius "a murderer and a villain" - does she understand what he means, specifically? Does she stop the "bloat king" Claudius from tempting her to bed after that scene? Shakespeare doesn't specify. It's up to your own reading of the text to make an argument for what Gertrude really knew.