My current laptop is broken, but the CPU is fine (Intel Core i7-720QM, 1.6 GHz).
Can I buy a new laptop and put this CPU in it ?
You technically can, but it's just not a good idea.
The i7-720QM uses Socket G1. A laptop with Socket G1 is going to be hard to find these days—modern laptops with socketed processors use the incompatible Socket G3. Besides, the Haswell processors in today's laptops are a lot faster and more efficient than the Nehalem processor you have, so don't bother.
We may be able to help you better if you can tell us why you want to reuse the old processor. It really doesn't make a lot of sense to do this. (Are you intending to connect your old hard drive instead? If this is the case, get a 2.5-inch hard drive enclosure instead.)