Hi Katie! Maybe someone else will know otherwise, but I don't think that there's an easy way to copy the quick filter settings across sheets. You could, of course, design a sheet the way you want it and then duplicate that sheet to retain all the formatting, but that only works when you first make the sheet.
You could potentially edit the XML contents of the TWB file to change all the quick filters, but that would be more of an advanced hack. Give that Tableau Workbook a Makeover with “Find and Replace Gone Wild!” – DataBlick