Hello Paul, the problem we have is it is too recent for most genealogy sites to record it fully, I can find reference to the birth in Dundalk, Louth, Anne Street is in the centre of town.
I see no other children born to a mother Travers listed but there are some born in Dundalk whese there is no mother listed, this is from Family Search (free site) and Ancestry subscription site. Details are not carried on these sites just bare information, Irish Genealogy a free Irish Government site is down currently but only has births to 1921 (they are using a 100 year rule for births so we are 8 years away) Roger said he looked for a marriage and there is no marrige to be found, however as the country divided in 1921 it is possible marriage took place in Armagh or Down two counties of Northern Ireland which border Louth and are still a sort of common area but different juristrictions.
As Travers is an unusual name I searched for Annie in the 1911 census and found a likely candidate in Dundalk not too far from Anne Street, in Wrightsons Place, Dundalk, actually near where a religious home of some kind as another query was here on it, children were sent there if a parent died or other misfortune. You can see the entry here http://www.census.nationalarchives.ie/ Search Annie Travers and age 15 in Louth, interestingly her father and mother were from Sligo and the children born in various counties and she was born in County Down now in Northern Ireland, a check in the 1901 census might show where they were then, the father was a carpenter. To check if she married in Northern Ireland you could look up PRONI here https://www.nidirect.gov.uk/proni ;
Her parents would have married in 1888 or thereabouts. You need her marriage certificate to confirm her fathers name and his for that matter.
There are few Mooneys in Dundalk in the census, I was surprised as it is a common enough name, there is a brother and sister in the 1911 census in Park Street and it and Annes Street run one into the other, however the two are a pharmaceutist and a teacher of English literature so some education implying some money, and they live in the pharmacy with workers and a servant.
The Irish census of 1926 will be released in 2026 so you may get some more info there, it hinges on finding the marriage certificate though and I will look again at the Irish Genealogy when it reappears online, it is possible it was not registered as the new state was in its formative years etc but there would be a church record, assuming RC as the donations indicate you could try writing to the parish church and see if they have a record.
There are some Louth private FB pages also, you can apply to join to see if there is anyone looking up the names, Travers is not a common name in Louth and only the one family in the census, Mooney is more common.
If you want any other webblinks let me know there is a good one on Louth but doubt it will assist you, it is here http://www.jbhall.clahs.ie/ ;
You need to locate the marriage certificate to proceed with any certainty.
I should be in Dundalk in the next month and if I see 62 Anne Street I will take a photo, you may find it on Google Street View.
Feel free to revert if you have any follow up queries.
Good Luck
Pat
