Hearthstone has finally added a major feature to their interface to better communicate to players about their quest progression when conceding games.

When pressing the concede button, you will now be given an additional prompt if you aren't going to make any progress on one of your quests to confirm that you are okay with this.

Quote From Out of Cards

Concede?
Progress for 1 or more of your quests will not be earned if you concede this early.

[Concede Now] [Keep playing]