QuickTime will not open .mov video

What is the best way to get this information?

This may depend on the codec used. Many will be recognized by the operating system—e.g., the Mac OS Finder "Info" window. Common third-party codecs not supported/opened by the QTX Player app can be determined using the "Inspector" or "Properties" windows of various media players such as VLC. In cases where the codec Profile and/or Level may be significant, the use of a dedicated media utility like MediaInfo would be recommended—e.g., especially for use with H.264/AVC or H.265/HEVC codecs. For example...

Finder "Info" Window

VLC Media Information Window Codec Tab

MediaInfo "Easy" View Window

And what do I do with it once I have it.

Compare the codec, Profile, and/or Level used to create your MOV files against the codec, Profile, and/or Level list supported by the media player your are attempting to use to view the MOV file. For example, an unencrypted WMV compressed video in an MOV file would not be compatible with QTX but would likely be compatible with VLC and may or may not be playback compatible with QT7 depending on your system's codec configuration or HEVC video should be compatible with High Sierra macOS QTX and iOS/tvOS v11 devices but not compatible with any other Mac operating systems or mobile devices. Basically, to play in the QTX Player app, the codec, Profile, and/or Level must be "natively" compatible with media structure built into the device operating system in use and only a very limited sub-set of modern codecs are so supported since the release of Mavericks and later operating systems.

I just exported a ~2 minute video I edited in Adobe Premiere and tried to watch it but found that QuickTime would not work. The "converting" bar fills up nearly all the way and then I get a message stating "QuickTime Player can't open 'Sequence 01_1.mov'. To see if additional software is available that will enable QuickTime Player to open the movie, click Tell Me More." There doesn't seem to be any real reason why QuickTime shouldn't work as it's a .mov file, and a short one at that. I've also exported videos from Premiere before and had no issues watching them in QuickTime.

Unfortunately, MOV merely refers to file container type and not the data format(s) contained within the file. An MOV file may contain any form of compressed data compatible with the system on which the file is created, whether or not the QTX Player can play the data. So the question becomes, "What form of compressed data did you place in the MOV file container—to include codec, profile and level as may be appropriate?"