No. As of 2nd August 2021 it is no longer a legal requirement to carry a Green Card.
You should always carry your V5 Vehicle Registration Document and a copy of your Certificate of Insurance, especially now Green Cards are no longer used.
You may also be required to have an International Driving Permit depending on where you are travelling to. This is available from the Post Office at a cost of £5.50 and must be applied for in person at a Post Office branch. You do not need an IDP to drive in the EU, Switzerland, Norway, Iceland, or Liechtenstein if you have a driving licence issued in the UK.
The UK Government have provided guidance around driving abroad on the following website;
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