Statutory birth, death and marriage registration (in some jurisdictions called Vital Records) only started in Ireland in 1864, save for non RC marriages which were recorded from 1845 onwards. So you probably won’t find birth, death or marriage certificates for this family. For earlier years you usually need to rely on church records, where they exist. You obviously need to know the precise denomination in order to search the correct records. Not all churches have records for that period and not all are on-line.
RC records are mostly on-line on the nli site:
https://www.nli.ie/en/family-history-introduction.aspx
For other denominations, the churches usually hold the originals but there are also copies in PRONI, the public record office, in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what record exists, parish by parish:
https://www.nidirect.gov.uk/publications/proni-guide-church-records
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
Looking at the 1831 census of Co Derry there were only 5 Greer/Grier households in the Coleraine area:
http://www.census.nationalarchives.ie/pages/1831/Londonderry/Coleraine/Aughadowy/Ballanacally_Mere/14/
http://www.census.nationalarchives.ie/pages/1831/Londonderry/Coleraine/Aughadowy/Ballymena/10/
http://www.census.nationalarchives.ie/pages/1831/Londonderry/Coleraine/Ballyrashane/Island/1/
http://www.census.nationalarchives.ie/pages/1831/Londonderry/Coleraine/Killowen/Killowen/107/
http://www.census.nationalarchives.ie/pages/1831/Londonderry/Coleraine/…
The 1831 census didn’t detail every member of the family. It just named the head of household, so no way of telling whether 16 year old John Greer was in any of the 5. However it gives you parishes and denominations to base a search in PRONI on.
Esther’s maiden name may have been Weir. That’s a common name in the area.