Yes QM2 is child friendly with a full childrens programme. You can see an example of the programme on the cunard.com website (does not show up on cunard.co.uk). Kind regard.
There is a nice kids club, but i dont know if you would wish to leave a 2 year old alone. the ship is child friendly, although i wouldnt take mine on this particular ship (i have 4 kids)i feel i would be looked at by some of the passengers!. the ship is far too civilized for that.