How can I check index fragmentation in the quickest way possible? Checking index fragmentation in my database seems unruly slow. Regardless if I use the DMV sys.dm_db_index_physical_stats (for a specific database, table, or even index) or if I use the SSMS Index Properties window to look at Fragmentation on a specific index, it takes a really long time. 

For example, using the Index Properties window will take upwards of 5 minutes to open up for a single index on my largest (~20 billion rows) table.

I do want to push to implement partitioning but until then I have to support an existing index maintenance job and I'm not sure how we can even check index fragmentation when one index on our heaviest table takes about 5 minutes to analyze. (Each of our tables has at least a few indexes.)

Here's a case where it took so long in the Index Properties window that I think it timed out and returned nothing in window:
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.