You can search birth records on the
Irish Genealogy website. It is run by the Irish government, and searches are free.
If you are not sure where she was born, it's possible a search may yield a few people with the same name. In that case, t would be helpful if you know your great-grandparents names. Sometimes those names appear in the search results. If not, you may have to open the scanned image of the birth ledger (as long as one is available). Then you can check for your great grandparents names. Obviously, if your grandmother's name was Mary Murphy, and her parents were Patrick Duggan and Margaret O'Connell, you should see those names on the birth record. Otherwise, it likely is not the right person.
I also first checked the
census records. I was not 100% sure of my great-parent's forenames. But I did know my great-grandmothers's maiden name. My grandfather had a lot of siblings, and I knew the names of several. I knew he was from Co. Clare. So when I found a census record for his family name, and a few of the children't names matched, I was certain I had the right one. That allowed me to verify my great-grandparent's forenames.