To begin with let me fix some notations. Let $\mathbb{T}=\{z\in \mathbb{C}:|z|=1\}$ with the normalized Lebesgue measure $\mu$. Let $C(\mathbb{T})$ denote the space of all continuous functions on $\mathbb{T}$. If $f\in C(\mathbb{T})$, we will say that $f\geq 0$ if $f(z)\geq 0$ for all $z\in\mathbb{T}$. Then Fejer-Riesz lemma [1] states that
Let $\varphi\in C(\mathbb{T})$ be given by $\varphi(z)=\sum_{n=-N}^N a_nz^n$ for $z\in \mathbb{T}$. If $\varphi\geq 0$, then there exists a polynomial $p\in C(\mathbb{T})$, with $p(z)=\sum_{n=0}^N p_n z^n$ such that $\varphi(z)=|p(z)|^2$ for all $z\in\mathbb{T}$. In other words, $\varphi$ factorizes as $\varphi=\bar pp$.
Now from the theory of Hardy spaces, we know that if $f\in L^{\infty}(\mathbb{T},\mu)$ and $g\in H^{\infty}$, then the Toeplitz operators corresponding to them, $T_f$ and $T_g$, respectively, obey the relation $T_fT_g=T_{fg}$ [2].
Also, if $f=\sum_{n=-\infty}^{\infty}c_nz^n\in L^{\infty}(\mathbb{T},\mu)$, then the formal Toeplitz matrix corresponding to $f$ relative to the canonical basis $\{z^n:n\in\mathbb{Z}\}$ is \begin{align} \begin{bmatrix} c_0 & c_{-1} & c_{-2} & \cdots \\ c_1 & c_0 & c_{-1} & \cdots \\ c_2 & c_1 & c_0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{bmatrix} \end{align}
Going back to the statement of Fejer-Riesz lemma, we see that $p\in H^{\infty}$ and $\bar p\in L^{\infty}(\mathbb{T},\mu)$, so that we can write $T_{\bar p}T_p=T_{\bar pp}$. Also corresponding to $\varphi$ we have $T_{\varphi}$. Then the relation $T_{\varphi}=T_{\bar p}T_p=T_p^*T_p$, when written in the matrix form yields \begin{align} \begin{bmatrix} a_0 & a_{-1} & a_{-2} & \cdots & a_{-N} \\ a_1 & a_0 & a_{-1} & \cdots & a_{-N+1} \\ a_2 & a_1 & a_0 & \cdots & a_{-N+2} \\ \vdots & \vdots & \vdots & \ddots \\ a_N & a_{N-1} & a_{N-2} & \cdots & a_0 \\ \end{bmatrix} = \begin{bmatrix} p_0 & 0 & 0 & \cdots & 0 \\ p_1 & p_0 & 0 & \cdots & 0 \\ p_2 & p_1 & p_0 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & 0 \\ p_N & p_{N-1} & p_{N-2} & \cdots & p_0 \end{bmatrix}^*\begin{bmatrix} p_0 & 0 & 0 & \cdots & 0 \\ p_1 & p_0 & 0 & \cdots & 0 \\ p_2 & p_1 & p_0 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & 0 \\ p_N & p_{N-1} & p_{N-2} & \cdots & p_0 \end{bmatrix}.\end{align}
If we focus on the main diagonal on both sides, we get equations \begin{align} a_0 &= |p_0|^2 + |p_1|^2 + ... + |p_N|^2, \\ a_0 &= |p_0|^2 + |p_1|^2 + ... + |p_{N-1}|^2, \\ &\vdots \\ a_0 &= |p_0|^2, \end{align} which imply that $p_1=...=p_N=0$. Consequently, the matrix corresponding to $T_p$ is diagonal, and hence we conclude that the matrix corresponding to $\varphi$ is also diagonal, which might not be true. So, I'm erring somewhere, but I am not able to see it. Can someone kindly shed some light?
P.S. - I haven't defined $H^{\infty}$ space or Hardy spaces. I can include if people feel it should be included.
References:
[1] Lemma 2.5 in Paulsen's Completely bounded maps and operator algebras,
[2] Proposition 7.5 in Douglas' Banach algebra techniques in operator theory.
The mistake is in your matrix forms for $T_\phi=T_p^*T_p$. Note that the corresponding matrices are semi-infinite as the operators apply to a general function from $H^2$ having all positive powers of $z$ up to infinity. Your finite matrices are $(N+1)\times (N+1)$ truncations of those matrices, so the factorization is not true. Just continue those blocks to the right and down to infinity respecting the Toeplitz structure.