I just checked this link and it's looks like FCPX have no problem to open it.
.mts is also called AVCHD.
I also check FCPX Documentation here : http://help.apple.com/finalcutpro/mac/10.0.6/#ver2833f855
And it say it can support AVCHD videos.
VLC also support it for watching the video and anyway Quicktime X is useless with many codecs.
Check again to import files in FCPX as i can't test i don't have any .mts files.
Hope this help.
You can read this also if you want to convert them : https://discussions.apple.com/thread/1412182?start=15&tstart=0
I will say it's better to use Handbrake, anyway it's what i use.
FCPX by default can directly open .mts files, that is no doubt.
To make your mts files playable on QuickTime, you must upgrade your Mac OS to Montain Lion version. you can check the detailed info. here:
If you dont want to upgrade you Mac OS X, you may also do so other task like conversion from mts to an QuickTime friendly file like mov, mp4, 3GP, etc.
check the possible solution: http://www.ilikemall.com/how-to/convert-mts-to-mov-mac.html
Hope it works for you.