Ternaquin would definitely be referring to Tirquin, which is a townland just SE of Omagh. It has been referred to as Terquin, Terriquin, Terraquin, Trioquin on many documents and im guessing that Ternaquin would have been an incorrect transcribing of "Terraquin"
The majority of my ancestors came from Tirquin and the neighbouring townlands Ballinamullan and Maine between about 1770-1900 as far as i can tell.