Yes you can apply for a national identification card. However, you must not only provide your Australian birth certificate but also your marriage certificate and divorce decree absolute. Most importantly, you must produce your Certificate of Citizenship or Residence granted by the Ministry of National Security of Trinidad and Tobago.
The status of Citizenship or Residence would have been granted as a result of marriage to a Trinidad and Tobago national, provided an application was made. However if you were divorced before acquiring citizenship or residence status you will be ineligible to apply.