Japanese Sparrowhawk breeds in the forested hills of the region. It's probably reasonable common but sightings are unpredictable. They are far more conspicuous on migration than in continuous mountain breeding habitat and it's even possible to see them in city parks or along major river systems.
Seeing one dash by in woodland is one thing, but finding them perched requires even more luck.
Eurasian Sparrowhawk can be equally difficult to find in summer but in winter they also move into open country and are far more easily seen. I'll often come across one at Ogura or around Lake Biwa, hunting along hedges, riverside trees or even across open fields.
A spring migrant on Mishima, Yamaguchi.
A typical winter bird sitting in riverside trees at Ogura.
Another island migrant, this time on Hegurajima in Ocober.
Goshawk is also a common winter hawk, often seen in open country, much as Eurasian Sparrowhawk, and they are frequent in the Ogura area. While Eurasian Sparrowhawk is something I often see on islands during migration, Goshawk seems less common out there.