How can I check index fragmentation in the quickest way possible?
It looks like the other suggestions so far all lead back to sys.dm_db_index_physical_stats

This is where you will have to look to get the index stats, but you could look to use the LIMITED option and restrict it to specific indexes at a time.

With the size of your table, I would want to know how quickly fragmentation occurs to help me confirm that the indexing (particularly the clustered index) is appropriate.

How big is your maintenance window?  Is it all consumed by the scanning part? 

One other aspect to investigate (you didn't mention which version you were on) is RESUMABLE ONLINE REBUILDS.  If you have the space then this might be an option to run as much as you can over multiple maintenance windows.