An infinitive is the word to followed by the base form of a verb. (See 48b.) Subjects of infinitives are an exception to the rule that subjects must be in the subjective case. The subject of an infinitive must be in the objective case. Objects of infinitives also are in the objective case.
The infinitive phrase whom to believe is the direct object of the verb know, and whom is the subject of the infinitive to believe.
note: In spoken English, who is often used when the correct whom sounds too stuffy. Although some readers will accept constructions like Who [not Whom] did Senator Boxer replace? in informal written English, it is safer to use whom in formal English.