I have been trying to use SyncLion but due to Google Voice I have not been able to successfully test it. Still I had one of my friends try it out and they say it works nicely. SyncLion allows you to automatically backup or archive your SMS to your IMAP capable email account like Gmail and many more. It does not seem to constantly run in the background so it looks like you need to run the app every once in awhile and synchronize. I cannot vouch for how well SyncLion works but it seems to be the only solution currently for BlackBerry 10.
Check out SyncLion for $2.99 in BlackBerry World