The show of hands in favour of the merger was taken at the society's annual general meeting (AGM) in Cardiff on 25 July. GHS chairman Dominic Cole said: "We had an extremely positive 'yes' vote, which has enabled us to go forward."
He explained that the next step in the process is when the AGT takes its vote on the merger at its annual meeting in September.
"If they are in favour of it then there is going to be a transition board formed. The transition board will then address various aspects (of the merger) such as its constitution and its branding.
"The aim is for the proposals to be available at a joint AGM next July in Newcastle. So the hope is that the AFG will vote positively. If we can go ahead then that process will start immediately."
Last year, the two associations agreed to set up a joint merger project board to investigate issues surrounding any possible merger. The stated aim was to "create a stronger, more efficient national organisation".
The groups will also be consultees for Government planning policy.