He seems unaware of the conflict between my father and me.
Literal
Father [and-と] my conflict [to-に], he [topic-は] noticed-not appearing [copula-です].
葛藤 (かっとう) means 'conflict/struggle' — a literary word. ~ようです expresses an inference based on observation: 'it appears that.' 気付いていない = 'hasn't noticed' (resulting state of not noticing).