First, make sure your AE is up to date. Someone with experience with this flavor of JVC quicktime will have to chime in to let you know if this is a common problem with this type of footage. 3rd party video codecs or codec packs installed on your system this may be causing the problem.
If you must transcode the footage there is no reason to have a loss in quality if you use the right workflow. DO NOT compress it, you render a DI (digital intermediate) using a lossless production format. The best option is to use a 10 bit or better codec. This will not drastically lower quality, it is the standard way to get compressed footage into the production pipeline so you can work on it without quality loss. Production formats that work well are Quicktime JPEG 2000 (only 8 bit but supports transparency), the Cuneiform 12bit format, and a bunch of others. It all depends on your system. You can use the Adobe Media Encoder to do the transcoding. Newer versions of the Media Encoder have the Cuneiform Codec available. Here's the AME from CC. The Uncompressed YUV 10bit is excellent but gives very large files. There are other options depending on your OS and system specifications.
Thank you for your reply! A little bummed out that I need to incorporate a new workflow with this type of footage - considering they are still MOV files. BUT your suggestion about transcoding did do the job just fine. I was able to successfully transcode my footage with [seemingly] no quality lost, and I can now continue forward into AE. So again, thank you very much. A big life saver you are.