It's helpful to use the correct titles when writing to the Governor-General or her spouse.
The Governor-General's full title is:
Her Excellency The Right Honourable Dame Cindy Kiro, GNZM, QSO
Governor-General of New Zealand
The title of the Governor-General's spouse is:
His Excellency Dr Richard Davies
If you are writing to the Governor-General, address the letter to:
Her Excellency The Right Honourable Dame Cindy Kiro
Governor-General of New Zealand
Government House
Private Bag 39995
Wellington Mail Centre
Lower Hutt 5045
You can write to His Excellency at the same address. In either case, start your letter with: Your Excellency.