No, you don’t need a Green Card to drive in the EU.
From 2 August 2021, the UK became part of the Green Card Free Circulation Area (GCFCA) again, which is made up of all 30 European Economic Area (EEA) countries, as well as Andorra, Bosnia & Herzegovina, Serbia, Switzerland, and the British Overseas Territory of Gibraltar. This means you no longer need to carry a Green Card when driving in these countries.