As the most recent event in Ireland the baptism of Maria c1844 would be the best starting point - if that could be located it would give a starting parish and denomination.
Parish records for most of the Catholic and Church of Ireland parishes in Dublin City, plus a few south County Catholic parishes are available to search on the IrishGenealogy website mentioned by Kaye in the earlier reply, but a search of this for Maria/Mary with parents John and Winifred around the likely date is not showing any matches.
It's possible that your Byrne family lived somewhere outside the city, i.e. County Dublin in an area not covered on IrishGenealogy. Images of most Irish Catholic Parishes are available on the National Library Website, and indexed on both Ancestry & FindMyPast, but nothing promising showing up in Co. Dublin there either. Plenty of Marie/Mary Byrnes with father's name John between 1838 and 1849 (over 100) but none with mother's first name Winifred or similar. Similarly nothing promising for a marriage of John & Winifred - only Byrne marriages where the bride's first name was Winifred or similar that I found that were within the likely timeframe were a Thomas Byrne in Booterstown (south east Co. Dublin) in 1839 and a Matthew Byrne in 1844 in St. Nicholas of Myra Dublin city.
Transcripts of some Co. Dublin Church of Ireland parishes are available on RootsIreland (pay-website) but nothing proming on their index in Co. Dublin for Maria either. It's possible they did live in the city but there's no record - some registers dont go back far enough, plus there are gaps in registers - faded or missing pages etc, in addition quite a few of Church of Ireland registers were lost during the Civil War.
What was the source which mentioned they come from Dublin ?
There might be a possible lead in Co. Wicklow, which is the county to the south of Co. Dublin, with a Mary Byrne bapt. in Glendalough Catholic Parish in 1843, parents are John Byrne and Winefrid Mason. There is also a marriage of a John Byrne and Winefred (no surname recorded) in the adjacent parish Rathdrum but the following year 1844.
Might birth or baptism records for that area of New York State be available for any of their children, or maybe marriage records ? if so these might give you Winifred's maiden surname which might help the search... and could confirm, or rule out, this possibility ...
p.s. I suspect that marriage is a different couple - there are several John Byrne / Winifred couples in Co. Wicklow around this time (The surname Byrne is most common in Co. Wicklow).