Rather than convert, you'd be better off capturing with a more current version of the program -- one that supports AVCHD video. Otherwise, you're in for a lot of headache. A converter isn't going to solve your problems.
BTW, when you say Premiere 6, do you mean Premiere 6 -- which is over 10 years old? (There was no Premiere Elements 6.)
Trust me on this: Capture with a Mac version of Premiere Elements 11 or 12 and your video will be perfectly compatible with any Mac program.
Welcome to the forum.
If you do mean Premiere 6 (prior to the PrPro days), please let us know, and a MOD can move your post to the Adobe Premiere CS 4 & Before (covers Premiere before PrPro, and not Elements) Forum.
I agree with Steve. PrE11 or 12 are very good with editing AVCHD clips. That is what HG20 generates. If you have to stick with the professional software (vs PrE, which is for hobbyist video editors), try Premiere Pro CC.