The exact linguistic composition of Ulster in the 19th century is not easy to recover because one of the languages of Ulster, Ulster Scots, was not at all considered in censuses taken during that time. Scots is closely related to English, but is considered a separate minority language. It has been spoken in Ulster since the late middle ages, but only by large portions of the population since the 17th century. Speakers of Ulster Scots would have been lumped in with English speakers in the censuses.

However, if what you really want to know is how many people spoke Irish, the census records would be a good place to start. The 1901 and 1911 censuses recorded whether a person could speak the Irish language. You could look up the census yourself and browse pages from the neighbourhood the novel is set in to get an idea of how many people in the area spoke Irish. (Most censuses are free to search on Familysearch.org.) However, in the 19th century, the Irish language was definitely not required learning by English-speaking incomers to Ulster. It's realistic that the character's father would not have had to learn Irish when he moved to Ulster. The Irish language received little to no insitutional support in Ulster until the 21st century, and so the language has suffered by comparison to the Republic where it received active support from the Irish Free State's government: https://halshs.archives-ouvertes.fr/halshs-01147770/document

In 19th century Ulster, however, the ability to speak Irish would have fallen along largely ethnic and religious lines. As Protestants, the characters in the novel would have socialized mainly with other English-speaking (or Scots-speaking) Protestants, and their government was handled entirely in English, so there would have been no need to learn Irish, even if there were Irish speakers living nearby.

It looks like there is some information about numbers of 19th century Irish speakers here: https://www.jstor.org/stable/pdf/25506114.pdf