The correct option is
C Whom
Option C: 'Whom' is a relative pronoun that relates a clause to the rest of the sentence. Here, it would relate the clause 'she told the whole story' to the main clause/main sentence 'She came across a gentleman'. The relative clause adds more information about the 'gentleman'.
Also, 'whom' is an object pronoun that receives the action 'she told the story'.
Hence option C is correct.
Option A: 'That' is not an object pronoun. It means that 'that' is used as a subject that performs an action, rather than receiving it.
Here, the pronoun receives the action, that someone 'she' is telling them a story.
Hence 'that' is not suitable.
Option B: Similarly, 'who' is a subject pronoun that would perform an action.
Option D: 'Which' refers to an object whereas the pronoun must refer to a person 'gentleman'.
Hence options A, B and D are incorrect.