So why didn't the Qartheen kill Dany and take her Dragons ?

[Parwan]

...

As to the original question, I think it's because not everyone in that world is instantly ready to escalate to open murder before trying other more socially and legally acceptable ways to get what they want.

Basically right. The Qartheen and a great many others in Esos see the dragons as either things of wonder or matters of casual interest. There is no sense of threat at all. Dany is greeted by an honor guard and makes a sort of grand entrance, with a parade down streets lined with Qartheen sight seers. Only the entrance isn't that grand. Dany feels like a rather grubby barbarian compared to the beauty of the city and its people. The Qartheen are interested, even somewhat entranced. Many people come from foreign cities to view these wondrous beasts. Most people, however, aren't either permanently impressed or frightened. Dany goes to make her case before the Pureborn. They are pretty much bored by the whole thing. Xaro apparently is the only one of the Thirteen who has an interest in Dany's little beasts, and he proclaims himself a man of peace. He is also a liar and a very slippery fellow. There's no conflict here though. He can be all these things.

Qarth is a very rich place. It is not in all ways a nice place, but that doesn't mean that its citizens will quickly opt for murder. They don't need to, and they don't have much interest in doing so. Four hundred years, right? That's a long time. The Qartheen don't seem that afraid of Valyrians. Besides, Daenerys is a Targaryen. The people of Esos would not be threatened by her, even at first. And she quickly makes her intention clear. She wants to sail to Westeros and re-establish her family's dynasty there. Where is the threat to Qarth in that?

The Warlocks, of course, have designs upon her. They make their try and fail.

[LifeRuiner]

snip

This is exactly how I read it. The Qartheen didn't have a reason to care more than they did. The warlocks were the only ones who had any clue about Dany and her dragons' future role on Essos, so they made the strongest grab at her/them. Also, it is unrealistic to make all characters and societies capable to go from welcome to murder in 0.2 seconds.

[Howling Mad]

Dany was origanlly invited to Quarth by the warlocks. The remainder of the city may have been afraid to cross the warlocks over the dragons or were waiting for the warlocks to act. By the time someone decided to act against Dany Lord Grandfather had arrived to ferry her escape.

[Josephxoxo]

Because Qarth is a civilized civilization. They aren't barbarians. They actually respect the last living Targaryen.