Step 1: Enable IMAP Access to your Gmail account
Before you can begin backing up your SMS to your Gmail account, you need to grant IMAP access to it. To begin, sign in to your Gmail account from your computer and navigate to Settings->Forwarding and POP/IMAP.Step 2: Download and configure SMS Backup+
The app which allows you to back up SMS to Gmail account is called SMS Backup+, and is available on the Google Play Store for free. Simply search for the app or download it using the link given down below:Download SMS Backup+ from the Google Play Store
Once the app is downloaded, open it up and tap on the “Connect” button to add your Gmail account to the app. A pop-up window will show you the Gmail account (or accounts) connected to your Android device, so tap on the one you want to back up your SMS to.Source: HowToGeek
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