It depends on the version of MacOS that is installed on your new MBP, but if it is Catalina, Big Sur or Monterey (as is likely), then no you cannot install LR6 on that system (the installer for LR6 is 32-bit code, whereas the later MacOS versions only support 64-bit apps).
What you should be able to do is migrate your apps and settings from the old Mac to the new MBP using the Migration Assistant....that should allow you to continue to run LR6 if the migration is successful, but be aware that if you ever need to re-install LR6 you won't be able to.
Depends on what your 'new' Mac can boot into. You can install IF your new Mac supports booting into an older OS like Majove that will allow you to install the 64 bit application with the 32-bit installer. But if not, you're out of luck. Plus, this old version isn't supported with newer hardware/OS.
Author “Color Management for Photographers" & "Photoshop CC Color Management/pluralsight"
If you can't install LR 6 on your new computer (as explained by others), the only way to continue to use your LR 6 catalog is to subscribe to the Photography Plan, which includes Lightroom Classic, which will be able to upgrade and then use your LR 6 catalogs. And then you can transfer your catalog and photos via these instructions.