No he wasn't. He was born in Linnett Lane. There is a significant link though. George Melly the merchant and politician of the nineteenth century lived in the Square ( on the corner at 90 Chatham Street), and his daughter the educational reformer Florence was born there in the same house. This George was the Jazz legend's great grandfather though. In Scouse Mouse, one of Melly's autobiographies he talks about visiting the house when growing up.
Even earlier, politician George's father, the traveller and naturalist Andre Melly ( who claimed he was descended from Christopher Columbus) lived at number 2 in 1827. And dined with the artist Aubadon during his Liverpool visits.
Where did the rumour it come from?
A surfeit of Georges, numerous important Mellys and not checking the dates.