You do not need to convert. Just import from the camera. iMovie will convert to Apple Intermediate Codec.
If you have Final Cut Pro X, you can import directly from the camera into ProRes 422. iMovie can edit ProRes 422 if FCP is installed, but if you have FCP, I would suggest editing in FCP.
If you attach your camera via USB make sure the camera is plugged in to wall power - not batteries only. Or if your camera has an SDHC card you can plug it in to a USB card reader and you don't need to worry about power. iMovie should see it automatically but if not, use File/Import From Camera.
Also, for iMovie to import from your camera, you need the entire card structure, including any empty folders. If you move the mts files off the card by themselves, iMovie cannot recognize them.
If that still does not work, let me know.