Hunting through the "Photo Investigator" kept by the State Records of NSW, the following three vintage photos lead me to suppose that this structure was used as a tramway building. Unable to find an image of the structure itself, I post images around CQ that appear to be similar. If you have a conclusion based on this photographic evidence - or other evidence - I would value hearing from you.
Thanks to Ann from Sydney Meandering, the puzzle is resolved.
This is the Harbour Masters cottage constructed in 1910 under the purview of the newly established Sydney Harbour Trust which was charged with rebuilding the Quay after the bubonic plague decimated in in the first decade of the new century. In 1988 it was converted to a Cafe and Oyster Bar in time for the Bicentennial celebrations.