Will Migration Assistant transfer Quickbooks 2016 to my new Mac?

Yes if you select it to migrate. However, are you sure that version will work on the macOS version on the new Mac?

If the past I have had problems with some apps successfully migrating.

A lot of times it is better to just migrate the user(s) vice apps and other things. The user(s) transfer dat and other things associated with the user but not app and some other things. You can then download and reinstall apps you need.

What system is your old MB running? If it's Mojave (10.14) or earlier you should run the free GO64 app to learn which of your apps will not run with Monterey.

If you're running Catalina or Big Sur you should still verify that your important apps will run with Monterey before upgrading to Monterey.

This say you need QB 2019 or later to be compatible with Monterey

https://quickbooks.intuit.com/learn-support/en-us/other-questions/quickbooks-desktop-mac-and-monterey/00/964288#