Irish was not my forte at school so I stand open to correction, but I think the name Quartertown in Irish is simply " Baile na Ceathrún". Muileann means mill, so 'Muileann Ḃaile Na Ceathrún' would mean Mill of Quatertown. Note omitting the prefix Muileann means there is no buailte/séimhiú on the B of Baile. And just to add while I think there may have been additions since I went to school, the letters J, K Q, V, W, X Y & Z, did not exist in the Irish alphabet.
For Irish place names see https://www.logainm.ie/en/s?txt=Quartertown
P.S. the H after the T in Ceathrún, should be omitted and replaced with a buailte as has been pointed out.