Email Alias set up

Quickmail allows users to send and receive emails using their email aliases in Gmail/G Suite.
Alternatively, you can also send emails on behalf of another email address (even non-gmail emails) as long as you can set the alternate email as "send mail as" address in Gmail. 

Once you've assigned an alias in G Suite, you can set it up as the "Send mail as:" address in your Gmail settings.
Go to Gmail Settings -> Accounts -> "Send mail as:".
You can also add a non-gmail email account if you want.

After that, go to Quickmail Settings -> Inboxes -> Settings -> fill out the "Replace Sent-From & Reply-to" with the Email Alias (or Send mail as:) email address. 

Note:  The originator (replaced email) of the email will still be visible when the email gets inspected. 
Setting up a non-gmail email account as "Send mail as" may cause problems with reply and bounce email detection. 
Replies and bounce notifications need to be in your gmail inbox for Quickmail to detect them. 
You can probably set-up email forwarding from the custom email account to send copies of incoming emails to your Gmail account.