Apparently, it’s because of some misunderstanding. From Deped CAR:
According to a legend, Apayao is derived from the combination from the Ilocano word “Apay” meaning “Why” and Tagalog word “Ayaw” meaning “Don’t like”, the legend tells of an Ilocano and Tagalog who accidentally met in one forest of the province. The Ilocano ask, “Apay adaka ditoy?” to which the Tagalog answered,”Ayaw” thinking that Ilocano is offering food. Not understanding each other, they fought. Men from a nearby village the two men fighting and hearing the word “Apay” and “Ayaw”. The place was then referred as Apay-ayaw by the villagers thus came the name Apayao.