The Queen and the Duke of Edinburgh Prince Philip met the elephant calf while touring a new two million pound home for nine Asian elephants on the eve of its opening to the public.
Elizabeth was born the day before the Queen's 90th birthday celebrations last year and named in her honour.
The Queen also fed seven-year-old Donnaa banana during a visit to the Whipsnade Zoo.
Both elephants live in the Centre for Elephant Care, a new custom-designed barn.
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Professor David Field, a director for the Zoological Society of London (ZSL), said the Queen was "absolutely delighted" to learn 72 stone (460kg) Elizabeth was named in her honour.
Assistant curator of elephants Lee Sambrook, who demonstrated how keepers clean the animals' feet and inspect their teeth, said the Queen was especially interested to hear about the round-the-clock care they receive and compared it to keeping horses.
Asian elephants are considered endangered due to habitat loss, poaching and human conflict.
The ZSL charity - which runs the zoo and of which the Queen is a patron - is involved in more than 50 conservation programmes to ensure elephants and humans can coexist peacefully.
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