I'm not sure which application you use, besides Gmail. Also important- you mean the text messages appear in real time, or is it more a planned backup?
Anyway, to get you started:
Try Email> Server Settings> Uncheck SYNC SMS, if you're Email is in sync with Gmail, maybe the messages travel via Email.
Or, when they're forwarded directly to Gmail, you could try
Gmail> Settings > Forwarding and POP/IMAP, and disable IMAP.
So if it's not an Email client thing, nor some Backup SMS+, there's also Hangouts. You can find the settings there. This'll help:
https://productforums.google.com/forum/m/#!msg/hangouts/zdcoK74i9qM/Nsc6DuZ4-vIJ.
Keep in mind that some settings can only be changed on your desktop...
Goodluck
Whispering Wempe (Swyped from my Galaxy SIII mini, via AC App)