would it be legal to sell a qt/c++ application (lgpl) without buying a license? And would doing the same but with pyqt(GPL) be legal? If it's not, should I explicitely tell the customer (It's an specific application for only a few customers) It's license, I don't think he/she would understand a thing. And I guess we will install the software ourselves on their computer, so Where I should put the source?
Of course you can sell it. GPL doesn't state that the software needs to be free (as in beer), but you need to give out the source code (or the possibility to obtain the source) as well as the binaries. For the last question, you don't need to put the source on the customer's computer, you can for instance give them a link to the source code on a website or revision control repository.
would it be legal to sell a qt/c++ application (lgpl) without buying a license? And would doing the same but with pyqt(GPL) be legal?
Yes and yes. The GPL doesn't prohibit to sell your software. The whole point of using the LGPL basically is that, in contrast to the GPL, you don't need to provide the source code of your software to the third party.
And I guess we will install the software ourselves on their computer, so Where I should put the source?
You don't need to put the source anywhere specific. You just have to provide your client a way to obtain the source, e. g. by asking you to send it to them on demand. You could also just give him the source code archive and you're done.
Just make sure you give him the license text in some way, e. g. bundle it with the software, show it on installation or make a menu item in the "Help" menu showing the license text. If your client doesn't care for it, that's fine.