AusLit suggest a first publication date of 1947. If you do a Google search for Mollie Quick (note spellings) you will find that she illustrated a number of children's books published in Sydney late 1940's.
http://www.austlit.edu.au/austlit/page/A48577
I'm afraid I can't help with who Mollie Quick was - it may well have been a pen name. The only reference I can find apart from as a book illustrator is an article in 1946 edition of Home Beautiful titled "Minimum Rural House."
http://members.iinet.net.au/~graybut/Home%20Beautiful%20index%201938-1971.pdf
These books can sell for considerable sums in USA, though you do have to be creative with how you list them on eBay (as you have discovered N***** is a "bleep" word.
Ten Little N***** Boys (Cover Title: Ten Little N******) - 1900 ended item
This was of course a much earlier book, but apparently copies like yours have achieved the $150 mark. (I am assuming it is in good condition inside and complete.)
