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The Quotes section there is actually pulling its fields from a related list, rather than a compact layout. You'll only be able to affect what is visible there by changing the page layout that the user is seeing. All of the fields and related lists in Lightning are based on whatever is displayed in the non-Lightning page layouts.
You should be able to click the gear and go to Edit Layout. When you click on the Quotes section there, it should show on the right side a list of the page layouts that are feeding into this lightning page. You can edit each of the page layouts from here, finding the Quotes related list and editing its columns. Be sure you add the desired column close to the left; the further to the left the column is, the higher it'll be in the Quotes section of the Lightning page.
