Hank here from Shopify.
What seems to be the issue, and what steps were you taking prior to reaching out to us here just so I know what stage you're at?
As Shopify only offers email forwarding, you have to have a separate account with Gsuite or Zoho mail to be able to reply to customers from your custom/chosen email. Has this been done within Gsuite?
Hopefully, that video will show you more of the process of what you are trying to achieve. But let me know if you are still having an issue.
You can also reach out to Google directly if you are noticing an issue within the Gsuite setup.
All the best,