E-invoicing by notified persons is mandated to supply goods or services or both to a registered person.
Thus, where the Government Department doesn’t have any registration under GST (i.e. not a ‘registered person’), e-invoicing doesn’t arise.
However, where the Govt. department has a GSTIN (as an entity supplying goods/services/ deducting TDS), the same has to be mentioned as the recipient GSTIN in the e-invoice.