Apparently the Whitmore Hotel, a little further west down Wright Street, which opened on March 23, 1839, was named the Queen’s Arms, and was under the management of one Thomas Maslin. It remained known as such until 1904 when it was renamed the Bushman’s Club Hotel.
This suggests that the present Queen’s Arms Hotel, which was licensed in 1851, opened with another name. We might conjecture that the present name was not taken before 1904? I am unable to find any further historical information about this hotel.
http://www.whitmorehotel.com/