Is it possible to use GitHub Desktop application to commit to Bitbucket repository?

If what you want is using Github App for pushing your code to Bitbucket what you need to do is going to "Settings" (in your local repo) and change your Primary remote repository (origin) to point to Bitbucket in your Github App. Then you can Sync Branch.

A .git repository is a .git repository. You can read your local repository with any client app. Clone the repo however you want, then open it in whatever software you want.

edit:

GitHub for Mac is optimized to work with GitHub remotes — but if you wish to use a non-GitHub remote, it will work just fine. Set the remote manually in the 'Settings' view and everything else should work as expected.

https://help.github.com/articles/can-i-have-multiple-git-remotes-non-github-remotes

Since you are in Mac. just open the Github app then drag and drop the folder into the app ;) Just make sure you have the git initialized and setup before dragging.