If not, how do I get the correct version?

No. I believe AVCHD support began with version 6.0.2. The latest version as of this posting is 6.0.5. You should update your FCP and your Quicktime. Believe it or not you are probably going to need to upgrade your iMovie as well (you'll be using that to import your files.).

How do I update them?

I don't know if the jump from 5.0 to 6.0 is an update or if you have to buy a new version. But go here, it'll get you started:

Apple - Support - Final Cut Pro (http://www.apple.com/support/finalcutpro/)