They have been engaged for exactly three and a half days when Barney tells Quinn about him and Robin. She brings it up, after sending them looks and gets the suspicion that he's cheating on her. What he then tells her is even worse.
He explains how they started out as friends and she seemed perfect. She didn't want a relationship, she played laser tag with him, she liked hard drinks and she was gorgeous. (Quinn wrinkled her nose at this, and he had to apologize six times before going on.) However, with their hatred of commitment and Robin's strange attraction to Ted, they remained friends instead of going into a relationship.
It wasn't until Robin's old boyfriend from Canada came back, that Barney realized he had feelings for her. (Quinn frowned in disgust at the mention that Robin was from Canada, and Barney did something he thought he would never, ever do. Defended the frozen wasteland.)
It was after the jerk came back and played with her, that they slept together and Barney discovered that his feelings went far beyond liking this girl. He actually imagined himself settling down with this girl, having kids, the whole nine yards. And it scared the shit out of him.
But nothing really happened, other than this strange, romantic tension that seemed to follow them everywhere. It made Barney fidgety and nervous and he went to sleep many nights dreaming of her. She seemed to like him too, but they both knew it was a bad idea. How could two people so similar be in a lasting relationship? (Quinn became very silent after this, leaving Barney to wonder if he fucked up again.)
They got together anyway. It was probably a horrible idea, considering all the indecision that stood in their way, but you couldn't keep such awesomeness apart. And, so it turned out, the awesomeness was what kept them apart. Or, that was what they would tell their friends, because they really didn't want to deal with all the emotional shit they felt. It wasn't their style.
After they broke up, Barney went back to his womanizing ways, though they were beginning to ebb. He knew in his heart that Robin was the only girl for him, and it killed him to see her go out with other guys and be happy, when some days he just wanted to die. (Quinn told him he was being over dramatic at this point. He wasn't dramatic, she was just insensitive.) So, he set out in search of a girl to get Robin out of his heart. He needed someone to help him get over her.
He met many women after that, all of them being the same. At least, that was how he saw it because he was frustrated and stubborn and didn't want to put in all the effort it took to make a good relationship work. This was the first time he admitted to that in the open.
Finally, he met a wonderful girl named Quinn who changed the way he looked at things. (He didn't say that one thing she didn't change was Robin and what she meant.) She was tough, funny, a stripper. Three things he loved. Three things that made an awesome person. Three things that made him propose to her.
He finished the story abruptly, because he realized about halfway through that he wasn't explaining their story to Quinn, he was informing her of it. He was telling her right off that she was not Robin Scherbatsky and she never would be. That he would always love someone else, no matter how rough and tumble Quinn could be. (He doesn't tell her this of course. After all, he needs to sleep on it and gather his thoughts. Maybe after he gets a good night sleep and an omelet he'll ask Quinn for that ring back.)
And when they go to bed that night, Quinn still silent and Barney lost in thought, he turned to the side, looking at Quinn's perfume. It was Daisy, by Marc Jacobs, the same that Robin wore. He grabbed it off the night stand, sprayed it in the air, inhaled, and smiled.