I did not know Councils got involved. I thought it was a state thing with mediators.
I see in Qld; you can bill the adjoining owner up to $300 a year on branches that overhang, that you have removed by a professional. But there are specifics, and the process is a hassle. Best to cut the ones you can reach and then lop them over. Be nice about it, and maybe they will help out and do it all so that they do not find a mess in their yard every week.