What is needed to play .flv files on Quicktime on OS X Yosemite?

The only plug-in I've ever been aware of that allows .flv files to play on Quicktime is Perian. You can still download it -- http://perian.org/ -- However, it has been discontinued and is no longer being supported. My recommendation is to use a different, more fully featured, player. You can't go wrong with VLC -- http://www.videolan.org/vlc/index.html -- it can handle just about any media format you can throw at it.

All you need here is a video converter that can convert FLV to MOV or other compatible format, as as to be played on your Mac OS. Actually VLC does not support converting videos on Mac, but you may try this video converter for Mac. Guess it works.