Can I put OS 10.14.6 (Mojave) on my new Mac Pro (2019) and will it run?

No, no Mac can run an OS that pre-dates the one it shipped with. The only way to do that would be with a virtual machine.

macjack is correct. Be aware although you can run Mojave as a virtual machine, certain features and apps won't work correctly.

bbdb wrote:

Can I put OS 10.14.6 (Mojave) on my new Mac Pro (2019) ...

That depends. Do you have its installer app?

... and will it run?

The answer is almost certainly no, but there's nothing preventing you from trying. If it doesn't boot you will need to restore its content from a backup or reinstall macOS.