They both are from Ireland, and this census gives an ballpark immigration year for both Hugh and Margaret.
name: Hugh Haggerty titles & terms: event: Census event date: 1900 event place: ED 104 Eyota Township, Olmsted, Minnesota, United States birth date: Oct 1828 birthplace: Ireland relationship to head of household: Head father's birthplace: Ireland mother's birthplace: Ireland race or color (standardized): White gender: Male marital status: Married years married: 46 estimated marriage year: 1854 mother how many children: number living children: immigration year: 1848 page: 3 sheet letter: B family number: 56 reference number: 78 film number: 1240778 digital folder number: 004120294 image number: 00731 Household Gender Age Birthplace headHugh HaggertyM 72 Ireland wifeMargaret HaggertyF 60 Ireland sonEdward HaggertyM 40 New York sonHugh HaggertyM 28 Minnesota
Their older son Edward was born about 1860 in New York, so maybe a birth or Catholic church baptism for him would give their home parish in Ireland?
Try searching with familysearch.org
Edward was born in March of 1860, if the census of 1900 in Olmsted is to be believed. Edward Haggerty United States Census, 1900 birth: Mar 1860 New York residence: 1900 ED 104 Eyota Township,? Olmsted,? Minnesota,? United States
There is a possible match for 1860 in New York,
name: Hugh Hagerty residence: , Ulster, New York ward: The Town Of Kingston age: 30 years estimated birth year: 1830 birthplace: Ireland gender: Male page: 406 family number: 2927 film number: 803870 dgs number: 4237097 image number: 00484 nara number: M653
Old Hugh's death certificate (at least the index that's online) seems to show his parents' names back in Ireland: https://familysearch.org/pal:/MM9.1.1/FDMX-5M4
name: Hugh Hagerty gender: Male burial date: burial place: death date: 29 Nov 1903 death place: Eyota, Olmsted, Minnesota age: 74 birth date: 1829 birthplace: Ireland occupation: Farmer race: White marital status: Married spouse's name: father's name: Edw. father's birthplace: Ireland mother's name: Catherine Carroll mother's birthplace: Ireland indexing project (batch) number: B02070-7 system origin: Minnesota-EASy source film number: 2117540
The 1900 census shows that Hugh came to America when he was about age 20 so it's hard to know if his parents came too, or if he came without them.