。
She quickly came to like him.
Literal
She [topic-は] quickly him [subject-が] liked-became.
The prescriptive standard for 好き: it's grammatically an adjective, so the liked entity takes が, not を. 彼女は彼が好きだ ('she likes him'). When 好き shifts into the change-of-state 好きになる ('come to like / fall for'), が remains the prescribed marker. In casual speech を is also heard with this construction (since people analyze 好きになる as a transitive verb), but が stays the safer choice in writing and formal speech.