The APK is the app. Installing an app on someone's phone is itself giving them the APK, so it's a nonsense to talk about installing the app without giving them the APK. It's analogous to asking if you can run a program on a Windows machine without giving them the EXE file.
If you instead want to install an app on someone's phone without them being able to copy the app to a different phone, the author of the app needs to use an anti-piracy library (a.k.a. "licensing service") to stop the app when it's run on an unauthorized phone. As programming questions are off-topic on this site, I won't discuss that further.