I suppose that anyone can call themselves anything other than their given name, and claim it as their "nickname". You could just as easily have a child named "Elizabeth" with the nickname of "Sally".
Most people, though, would not automatically associate the name "Lilly" with "Elizabeth" -- certainly not as easily as the nicknames "Liz" and Lizzie" are associated with Elizabeth.
If you're talking about naming a child "Elizabeth" and then calling her "Lilly", she'll probably grow up having to explain to people all the time why her legal name differs from the name by which she is known.
Why don't you simply name your child "Lilly Elizabeth"? Or even "Lily Elizabeth", as the flower "lily" is correctly spelled with one "l".
Even "Lilibeth" or "Lilybeth" are pretty names for a girl.