Potential discussion topics for this year's synod include divorce, adoption, caring for the elderly and the acceptance of gay people in the church.
Mrs Cole said the belief that gay people are "intrinsically disturbed" was harmful and alienated New Zealanders from the Catholic Church and its teachings.
"We're a welcoming church but sometimes the way the church presents that is difficult," she said.
"We have a God of love and we should have a church of love."
Fellow delegate Archbishop John Dew said economic challenges and migration were also up for consideration at the three-week-long conference.
This is the Archbishop's sixth synod. He said he was pleased to be making a new contribution to the discussions.
"Everyone who is invited has the opportunity to give a presentation. Mine will probably be about the church walking with families and how we can support them."
Communications advisor to New Zealand Catholic Bishops Simone Olsen said it was important for isolated geographic regions like New Zealand to be represented.
"Pope Francis has always talked about going out to the margins and the periphery.
"I think in this case he has done that by including countries like Tonga and other Pacific nations that are not traditionally Catholic," Mrs Olsen said.
The delegation will not have a scheduled audience with Pope Francis but Mrs Cole is hopeful she will meet her "role model" during the event.
"I have huge admiration for his pastoral approach of love and acceptance," she said.
"I have so many non-Catholic friends who say what a good Pope and good person he is. I would just like to tell him that and give him that affirmation."
Bishop Charles Drennan and Dr John Kleinsman will also be in attendance.
