If your application isn't open-source, then it depends on how it is using calibre.
If it is just streamlining the process of installing useful software, then this presumably applies:
https://www.gnu.org/licenses/gpl-faq...ompatInstaller
If your application uses calibre during the course of its function, then you can only do that if it "communicates at arms length":
https://www.gnu.org/licenses/gpl-faq...prietarySystem
You would also have to make it clear that you are using calibre in that manner.
Without knowing more about what your application does, I don't think we can hazard a guess as to whether that applies or not.