Thank you Patricia and Geraldine for your research. It is very helpful. It is difficult to tell if the baptismal record you posted is for Martin. It does look like a listing for godparents but I am not sure. I am speculating on Martin's birth year. In the mortality schedule for the 1850 U.S. census, he is listed as dying in January at the age of 28. This led me to believe he would have turned 29 sometime in 1850 making him born in 1821. Since there is no way of knowing if he was actually 28 in January, 1850, there is a chance he could have been born 1819-1824?
I came to focus on Dunkerrin Parish as the location of Martin's birth because of the dna match on Ancestry. The dna match had a Robert William Maher in his family tree that I was able to show was the son of Roger "Roddy" Maher and the grandson of Thomas Maher from Moneygal, Dunkerrin Parish. I believe that Martin is connected to Thomas but I haven't found the documentation. Also, how likely is it that Thomas had his family in Dunkerrin but that he and Martin were born elsewhere in Ireland? How far would tenant farm families move within Ireland in the first half of the 19th century? Should I widen my search out from Dunkerrin?
Another piece of the puzzle might be a William Maher born around 1800. He shows up in the 1840 U.S. census and I think I have been able to connect him to a baptismal record for a James Maher in 1835 in Moneygall. He could be a much older brother/uncle to Martin. I am less inclined to think he is the missing father but Martin does not seem to have resided with him according to the 1840 census. The census at that time only listed heads of households but it doen't show a male of Martin's age living in this household. Martin may have still come with him to America.
Thank you for all your help. Family research often seems like a wild goose chase but I do like it!