Hello Johnny,
Not as far as I'm aware. Jays are still a legal quarry species but many same as myself tend not to shoot them. They are a member of the Corvid family rooks, crows, jackdaws, magpies etc probably why they're included but I'm not sure. If you look under WML Gen- L05 on the Natural England website it says the following :-
1. The purposes for which this licence is granted are preventing the spread of disease and preventing
serious damage to livestock, foodstuffs for livestock, crops, vegetables, fruit, growing timber, fisheries
or inland waters.
2. Subject to the terms and conditions below, and for the purposes set out in paragraph (1) above, this
licence permits:
(i) any authorised person to kill or take any of the wild birds listed in this subparagraph, to take,
damage or destroy their nests or to take or destroy their eggs:
Greater Canada Goose Branta canadensis
Crow Corvus corone
Dove, Collared Streptopelia decaocto
Gull, Great Black-backed Larus marinus
Gull, Lesser Black-backed Larus fuscus
Gull, Herring Larus argentatus
Jackdaw Corvus monedula
Jay Garrulus glandarius
Magpie Pica pica
Pigeon, Feral Columba livia
Rook Corvus frugilegus
Woodpigeon Columba palumbus
Hope this helps
Regards