will final cut pro handle MTS

As individual files, NO.

Import from Camera - most likely, YES.

Also, you can import from an archive on disk - make sure to copy the entire folder structure from the camera to disk, not just the individual files.

If you need to ingest individual mts files, then you'll need to use an utility like ClipWrap or MPEG Streamclip to create a Quicktime file. Clipwrap can do it without recompressing the video (just "wrapping it" in a mov container), but it is not free.