The GPL requires you to only provide source code whenever you provide binaries. You may opt to provide neither of these. If you provide the customer with the binaries, GPL requires you to distribute the source to the customer as well, but not to the whole world.
However, recent versions of QT are licensed using many licenses from which you can choose the one you use. The most relevant to you is probably LGPL 3.0. LGPL 3.0 requires you to distribute only the object code if using static linking. In your case, you do not need to even distribute the object code, as you are using libraries that can be installed from repositories. Presumably this means dynamic linking.
However, I am somewhat surprised that you do not want to provide the source code to the customer. If the application is developed specifically for this customer, usually it is considered good practice to provide the customer also with the source code.