You can receive support requests at an external email address, and forward it to your Zendesk account.
To do this, you need to configure email forwarding in the external email account, not in Zendesk. Exactly how this is done depends on the email provider you're using. A number of email providers allow you to create email forwarding rules that allow you to select the incoming mail that should be forwarded to your Zendesk account.
Follow the steps for your email provider to set up forwarding.
After you set up forwarding, you need to add your external email address as a support address in Zendesk. See Adding support addresses for users to submit tickets. When you add your support address, your email will be verified and you'll know whether you've set up email forwarding properly.
Optionally, you can edit or create an SPF record to verify that Zendesk can send email on behalf of your email server. See Setting up SPF for Zendesk to send email on behalf of your email domain.