Hi,
From your description, my understanding is that you have question about search REST API.
You could not get related results with search REST API. If you want to get related result by using REST API to get related result with the list name. you could refer to this article:
http://sharepoint.stackexchange.com/questions/119833/how-to-use-sharepoint-2013-rest-service-to-retrieve-data-from-lookup-field
http://sharepoint.stackexchange.com/questions/91682/help-with-getting-the-value-of-a-look-up-field.
In my case, I get the list name with this URL:
http://sp/sites/sharepoint2013/_api/web/lists/getByTitle('listBBBB')/items?$select=lookup/Title&$expand=lookup
you could refer to this screenshot below:

Then I get related result with URL:
http://sp/sites/sharepoint2013/_api/web/lists/getbytitle('ListA')/items
You could learn more about search REST API by referring to this article:
SharePoint Search REST API overview
https://msdn.microsoft.com/en-us/library/office/jj163876.aspx.
In addition, I recommend that you could accomplish your requirement with C# code, you could refer to this article:
http://sharepoint.stackexchange.com/questions/90623/how-to-find-lookup-field-list-using-programmatically.
Best Regards