Before a cat or dog can be imported into New Zealand from Belgium, an import health permit must be obtained from the Ministry of Primary Industry's (external link) quarantine service. The Import Health Standards outline the length of any quarantine period required. 

The Golden Pages (external link) or your local veterinary surgeon are useful sources to find companies specialising in the transportation of domestic pets.
Application forms for an import health permit can be obtained from the New Zealand Customs Service in Brussels:

Customs Service
New Zealand Embassy
Avenue des Nerviens 9 - 31, level 7
1040 Brussels
BELGIUM

Tel: + 32 2 550 1218
Fax: +32 2 513 4856

Other enquiries should be directed to:

Import Management Section
Animal Biosecurity Group
PO Box 2526
Wellington
Tel: +64 4 498 9625
Fax: +64 4 474 4132