Fasty Its not that hard to figure out. "Win one Quick Match" is the same as "Win one game" and "Win the game" just that it can only be done in a quick match and not a custom game—it will also say (quick match) at the end of the achievement description if you can only get it in a quick match otherwise you can do them in either a quick match or a custom match. It refers to winning the whole game.
e.g. the Curahee achievement says "Party up and win 1 quick match with friends." That means you have have to win an entire game with your friend.
"Win the round" is exactly what it says it is. You need to win one of the rounds in the game.
eg. the Touch Down achievement says "In Broken Arrow, survive over 2 min 30 sec in one round." That means you have to last that amount of time in a single round. It doesn't matter if you lose the entire game or if you even lose that round. You just have to survive for that long in one of the rounds.