Hi,
Like another person posted, the Blue Moon Skating Rink (roller skating I believe), was near where the GoMA and Art Gallery are based now in South Brisbane.
It may be completely different from that you seek though.

My mother visited it on weekends as a kid. The owner in the circa late '30's/early 40's, was 'Pop Flanaghan' (Flannagan?/Flannigan?).

Maybe you can use this as a basis for searching through Archives at the State Library or Archives/Council etc?

Good luck and hope it helps you to trace back to your family member.
Lisa