。
She struck up a conversation with the elderly lady.
Literal
She [topic-は] that elderly-lady [with-と] conversation [object-を] began.
老婦人 is a respectful, slightly formal word for an elderly woman — more dignified than おばあさん, with the kanji 婦人 carrying connotations of refinement. The と here marks the conversational partner (comitative と), distinct from the quotative と used for indirect speech. 話を始める ('begin a conversation/story') is a useful set collocation: 話 covers anything from a brief exchange to a long anecdote, and 始める supplies the inceptive sense.