No, this sentence is not a verb phrase, because the subject is not part of the verb phrase here.
Here's why. The subject is "I," the verb is "believed," and everything following the verb ("every word he said") forms the object of the verb. By definition, a verb phrase is one verb + its various objects or modifiers. Here, "every word he said" operates as one single object (it's not just one word, it's EVERY word, and it's not just every word, it's every word HE said). But the subject is separate from the verb phrase, so the entire sentence is not a verb phrase (it's a subject + a verb phrase).