Northern Ireland Secretary Owen Paterson met Mr McGuinness this morning and added that he was relaxed.
He said the Deputy First Minister greeted the Queen in Irish and their meeting was cordial.
It had obviously gone very well.This will move Northern Ireland on to a whole new plane. After all the trauma of Northern Ireland, everyone is looking forward.It is about a shared future, not a shared-out future.
None of this could have happened a few years ago so it is all looking to the future.
He said it was absolutely appropriate that when the Queen visits parts of the UK, she meets local politicians, democratically elected, pursuing their democratic political goals by peaceful means.
Mr Paterson added that it built on the success of the Queen's visit to the Republic of Ireland last year.
In an historic encounter, the Queen has shaken the hand of Sinn Fein's Martin McGuinness.
Today's handshake between the Queen and former IRA commander Martin McGuinness is a historic gesture of peace. Here are some others.
The hands of history? The Queen and former IRA Commander Martin McGuinness will make a remarkable gesture for peace today.