Export Related Records in Salesforce

Using Dataloader - NO

Data loader allows us to extract a single object data.

Use Chrome Extension named Salesforce Inspector, Workbench etc. - YES

Refer the following for details-

https://developer.salesforce.com/docs/atlas.en-us.lightning.meta/lightning/inspector_intro.htm

https://chrome.google.com/webstore/detail/salesforce-inspector/aodjmnfhjibkcdimpodiifdjnnncaafh?hl=en

Queries like the following can help-

child to parent -

SELECT Id, Name, AccountId, Account.Name FROM Opportunity

parent to child -

Use subqueries like SELECT Id, Name, (SELECT Id, Name FROM Opportunities) FROM Account

The Salesforce Tutorial