Do you have the manifest of the ship he came over on or do you know what port he came through? A ship manifest may show who he came with, if anybody.
I am unable to find anyone named Dan Quigley born or married in County Cork in Ireland civil registration around this time period. There IS a Dan Quigley born 1870 in the Stranorlar registration district - that is in Donegal. https://familysearch.org/pal:/MM9.1.1/FYYL-DNB
If you broaden the date range when searching for births, you'll find more.
Perhaps your ancestor went to Cork to study, and left from there?