The given blank is reported speech of an action in simple past tense.
"Walk" is option A is correct as the verb in reported speech of simple past tense takes the past perfect form. "I'd" is the contraction of "I had" and the verb will take the past perfect form "walk".
"Walking" in option B is incorrect as the gerund "walking"is in perfect continuous form and does not satisfy the rule of conversion.
"Walked" in option C is in simple past tense and is incorrect as it does not follow the correct rule of conversion.
"To walk" is incorrect as the transitive verb form "to + verb" is incorrect in the context.
Thus, option A is the correct answer.