トラベラーチェックの換金は、条件付きというのがNZ大手銀行の対応だと思います。
私が知っているのは、ASBとANZですが:
ASB
Travellers cheques, including those issued by Commonwealth Bank of Australia and American Express, can be exchanged for cash or deposited in to your ASB account. Conditions may apply so call 0800 803 804 or visit any ASB branch to check.
ANZ
We no longer accept travellers’ cheques, unless they were issued by ANZ New Zealand and are deposited into an ANZ New Zealand bank account.
まずトピ主さんが口座を持っている銀行に問い合わせたほうがいいと思います。