I'm afraid the XRMToolbox plugin doesn't support it yet.
Ref: https://github.com/MscrmTools/MscrmTools.PortalCodeEditor/issues/13
But there is no stopping you from creating your own pipeline - at the end of the day portal code is just bunch of Crm entities. Part of Crm SDK is configuration migration tool - last version is here:
https://www.nuget.org/packages/Microsoft.CrmSdk.XrmTooling.ConfigurationMigration.Wpf
So the idea is:
1) Get this tool
2) Define entities you want to backup & create schema xml file for them. I think you'd want adx_webpage, adx_webfile, adx_pagetemplate (and all attributes from them)
3) Export data using this schema - this exports them to .zip package that contains simple structure (schema file and data file); so you can unzip it and store in your git branch (pull)
4) For push zip this file and again use configuration migration tool to import the data
This gives you also an opportunity to have separate dev version of portal code and production version of portal code (which is always a good thing).