First of all, players such as VLC should be able to play the files just fine. Usually there's no need to convert. Just install VLC and use that instead of QuickTime.
Chances are the MTS files (MPEG Transport Stream) contain video and audio data that can be natively wrapped in a more widely supported container such as MP4, MOV, or MKV.
The free, cross-platform FFmpeg can change the container on the fly, without re-encoding your video. This means your quality will stay the same, and the conversion process will take a few seconds.
Download a static build of FFmpeg for OS X from their download page. Extract the archive and locate the FFmpeg binary, e.g.
~/Downloads/ffmpeg. Now, open up your Terminal.app from
/Applications/Utilities, and you can use the FFmpeg tool to convert video.
~/Downloads/ffmpeg -i input.mts -c copy output.mp4
Here, substitute the path to the input and output files to wherever your files are / should be.
This will copy both video and audio streams (
-c copy) and write them to
output.mp4. This file should be playable in QuickTime, and if it's not, you might need to install Perian, which is a codec library for OS X (VLC comes with its own codecs).
Finally, if you want to reduce the size of the MTS files, you could re-encode them, e.g. like this:
ffmpeg -i input.mts -c:v libx264 -preset:v slow -profile:v high -crf 23 \
-c:a aac -strict experimental -b:a 192K output.mp4
Here, the CRF value sets the quality. Lower means better, and sane values are from 18 to 28, depending on what you want. You can stop the processing with Q and take a look at the result file to inspect the quality. See the x264 Encoding Guide for more info and our FFmpeg blog entry for a general overview.