NO! My character had sex with Cullen, then fell asleep. That is not the bad part. The bad part is: She got woken up to a squirrel dropping an acorn on her head. THEN her face felt wet, upon touching it, a damn bird decided to drop down a "present" on her face.
Cullen really needs a new roof and a new floor.
When I'm running around Skyhold I often look at some of the other rooms that are in a better structural state and wonder why he didn't chose one of those instead lol I tend to just imagine that after all the years of being in the Circles, he prefers the illusion of open spaces, hence he has no roof and can sleep under the stars. My quizzy couldn't sleep in his room though, first night was a one off lol bet there are loads of bugs, spiders and midges
They move their relationship to her room for any "alone" time afterwards 
He does have quarters.
That's what he tells the Inquisitor when she expresses horror at his sleeping arrangements ("You can't seriously sleep here, Cullen, there's a bloody tree in the roof!"). And it's true; Josephine gave him quarters in the north wing, same as herself and Leliana and the rest of the inner circle (sans Sera, because she prefers the tavern, and Cole, because he doesn't sleep.)
It's just that he doesn't sleep there.
Oh he'd tried, the first week or so after Josephine assigned them to him, but it hadn't taken long for them to feel suffocating. The thick stone walls, the lone, narrow window, the lack of air; it was too much like his old templar quarters, and his sleep had suffered as a result. He'd jerk awake in the middle of the night from dreadful dreams and it would take far too long to remember that this was Skyhold, not Kinloch, not the Gallows; and there were no abominations outside to torment him or atrocities to follow him with accusing eyes. He could never go back to sleep, afterwards.
Bunking down above his office didn't exactly make the nightmares go away, but the cold, sharp air and the sound of battlement patrols felt like Haven and reminded him he was a soldier now, not a templar. It made the divide between waking and dreaming sharper. He slept easier.
When he eventually moved into her tower, he slept even better again.