Medieval travelers to the Holy Land saw crocodiles so representations of dragons on pub signs and in heraldry might have come from these exaggerated descriptions.
Pictured above is the Green Dragon in Marlborough and is one of many countrywide. A green dragon appears on the crest of the 1st Earl of Pembroke, a high-ranking Tudor nobleman whose wife was sister to Queen Catherine Parr. Two green dragon-like creatures also support the crest of the Worshipful Company of Plaisterers (plasterers).
To alchemists the green dragon is the universal spirit present in all things, not a mythical creature. Alchemy, a forerunner of modern chemistry, was considered cutting-edge science in the 16th Century; not only seeking to turn base metal into gold but also to purify and transform humankind. As early books on alchemy are recorded as being sold from the Green Dragon in St Pauls Churchyard, its fair to assume there might be some connection between the pub and the science (unless it was a very clever marketing ploy in 1652).
The most famous Green Dragon has to be the Hobbit alehouse in the Shire in Tolkiens Lord of the Rings. The Green Dragon Tavern in Boston was a favorite haunt of Paul Revere and the Sons of Liberty, and the headquarters of the leaders of the American Revolution.