As with Romania, I must extend apologies on behalf of Ireland to all my friends (and in this case my relatives) from a particular country. By far the most widely owned book set in Slovakia on both Goodreads and LibraryThing is a novel by an Irish writer exploring the life story of a Roma poet through the second world war and Communist Czechoslovakia.It's got lots of good reviews, and I guess I will look it up myself one of these days. It is:
Zoli, by Colum McCann
The book most often tagged "Slovakia" on LibraryThing is another Holocaust memoir, written for the YA audience, and published in 1999. It is:
Katarina, by Kathryn Winter
Tho most-tagged as "Slovakia" on GoodReads is yet another Holocaust memoir, though with quite a strong concentration on pre-war Bratislava. Published as recently as 2013, it is:
The Luck of the Weissensteiners, by Christoph Fischer
The best-known novel by an actual Slovak which is actually set in Slovakia is a crime novel whose story unfolds around the fall of Communism and the break-up of the former Czechoslovakia. It is::
Rivers of Babylon (originally written in Slovak despite English title), by Peter Pišťanek
I'm also going to give a shout out to my friend Rick's account of his father's flight from the 1968 Russian invasion, and his own rediscovery of Slovakia as an adult. (Hi, Rick!)
A Country Lost, Then Found, by Rick Zedník
I am intrigued also by the fantasy novel Vládce vlků / Master of Wolves, by Juraj Červenák, but I suspect it fails the "set in Slovakia" test.