score:4
I believe you're misunderstanding something here. According to the review,
The category “Word of God” is larger than Scripture because Scripture “does not exhaust the word of God” (p. 47). Therefore Frame defines “word of God” as “God himself, understood as communicator” and “the sum total of his free communications with his creatures” (p. 49).
Thus, the Bible is not (the entirety of) God's Word. Frame's definition of the Word as God recollects John 1:1:
In the beginning was the Word, and the Word was with God, and the Word was God.
Aquinas describes Christ, the second person of the Trinity, as "the Word". There's a long history, then, of using "the Word of God" as a way of talking about particular aspects of God; God in a particular relation to humans. But this is not (or at least not necessarily) to say that "the Word of God" is in fact, or even in Frame's perspective, anything like a "fourth Person" to the Trinity.
Further than that we can't really go based on the link you gave; this is, after all, only a book review, not a comprehensive summary.
Upvote:4
It's important to understand that the Bible is a man made thing--it's the name for a collection of 66 writings created throughout history that are recognized as the inspired word of God.
John Frame refers to the Bible as the Word of God, but what he really means is that the words contained within the Bible are the inspired words of God.
So the re-frame your question, could we say that God exists as the Father, Son, Holy Spirit, and the Word? That would be a misunderstanding as well:
John 1:1,2,14 ESV In the beginning was the Word, and the Word was with God, and the Word was God. He was in the beginning with God ... And the Word became flesh and dwelt among us, and we have seen his glory, glory as of the only Son from the Father, full of grace and truth.
Those that believe in the Trinitarian nature of God believe that Jesus, the son, is also the Word. So saying that the Trinity is the Father, the Son, and the Holy Spirit is true. And saying that the Trinity is the Father, the Word, and Holy Spirit is also true.