I understand that if a morph card is on the battlefield at the end of the game, I have to reveal it to my opponent, but what if my opponent responds to me casting the morph with a counterspell. If I concede with the morph still on the stack, do I have to reveal it?
Yes, you must reveal the card to all players.
707.9. If a face-down permanent moves from the battlefield to any other zone, its owner must reveal it to all players as he or she moves it. If a face-down spell moves from the stack to any zone other than the battlefield, its owner must reveal it to all players as he or she moves it. If a player leaves the game, all face-down permanents and spells owned by that player must be revealed to all players. At the end of each game, all face-down permanents and spells must be revealed to all players.
The other players must be able to check that you were allowed to play that card face-down. Otherwise, you could cheat by playing a non-morph card as a morph creature and never be forced to reveal that card.