How can I differentiate between Quote ID fields when I filter for Opportunity search?

There's only one field in the Opportunity object that's named Quote ID. However, the predefined Add fields feature displays the Quote ID field as a filter to search for opportunities based on the External Quote Number attribute of their associated quotes.

This results in the Quote ID field being available in a few predefined saved searches and can't be modified. Your administrator can create a custom Quote ID field by using a different name so that you can easily differentiate the fields to suit your requirements.

Note that if similar scenarios arise when searching across other objects where identical custom field names and predefined field names display in your search results, then you can request your administrator to create a custom field with a label that suits your specific business requirement.