How can I check email relay is working or not?

To see that your email relay is working, send a message from Salesforce and in the recipient's inbox, select the message and display the headers. For example, using gmail, you would use the "Show Original" context menu command

You can see the chain of email servers involved in sending the message by looking for the Received: entries. An email relay properly configured will have lines that look like this (Here, GSuite is the email relay)

    Received: from smtp02-ph2-sp1.mta.salesforce.com (smtp02-ph2-sp1.mta.salesforce.com. [13.110.6.225]) /*the email relay server ==> */ by smtp-relay.gmail.com with ESMTPS id f2sm207794pjs.0.2020.11.02.14.49.41 for  (version=TLS1_2 cipher=ECDHE-ECDSA-AES128-GCM-SHA256 bits=128/128); Mon, 02 Nov 2020 14:49:42 -0800 (PST) /* NOTE! */ X-Relaying-Domain: example.com Return-Path:  Authentication-Results: mx3-ph2-sp1.mta.salesforce.com x-tls.subject="/C=US/ST=California/L=San Francisco/O=salesforce.com, inc./OU=0:app;1:ph2;2:ph2-sp1;3:cs43;4:prod/CN=cs43-app2-3-ph2.ops.sfdc.net"; auth=pass (cipher=ECDHE-RSA-AES256-GCM-SHA384)

Common errors

You forgot to configure and activate a Sender Domain (even if simply *) in the Email Relay's Domain Filters.

You didn't whitelist the Salesforce IP ranges in your Mail Server

Note: Email relay is the process of transmitting an email message from one server to another.