If you mean all of the threaded text, then no. The only way you could do this is to make the contact a BBM contact and then your text messages become BBM messages which can be copied, emailed, saved, etc. The other person does not need to have BBM. You go to BBM and click on 'Invite Contact' -> 'Add a text messaging contact' and you are all done. Even existing SMS threaded messages will show up.
I was wrong. It just has the threaded SMS show up in BBM, but it is still like a threaded SMS in that you can't send the whole thing in an email like you can a BBM chat. Sorry.