4. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{2}=1, a_{n+2}=a_{n+1}+a_{n}, n \in \mathbf{N}^{*}$, then
A. There exist only a finite number of positive integers $m$, such that $a_{m}$ and $a_{m+1}$ are coprime
B. There exists at least one positive integer $m$, such that $a_{m+2} \cdot a_{m}=a_{m+1}^{2}$
C. There exist infinitely many positive integers $p$ and $q$, such that $a_{p} a_{q}-1$ is a perfect square
D. There exist infinitely many pairs of positive integers $(m, n)$, such that $a_{m} \mid\left(a_{n}^{2}+1\right)$, and $a_{n} \mid\left(a_{m}^{2}+1\right)$
For option $A$, $\left(a_{m+1}, a_{m+2}\right)=\left(a_{m+1}, a_{m+1}+a_{n}\right)=\left(a_{m+1}, a_{m}\right)=1$, incorrect;
For option $B$, when $m$ is odd, $a_{m+2} \cdot a_{m}=a_{m+1}^{2}+1$;
when $m$ is even, $a_{m+2} \cdot a_{m}=a_{m+1}^{2}-1$, incorrect;
For option $C$, when $m$ is odd, $a_{m+2} \cdot a_{m}-1=a_{m+1}^{2}$, correct;
For option $D$, when $m$ is odd, $a_{m} \mid\left(a_{m+2}^{2}+1\right)$, and $a_{m+2} \mid\left(a_{m}^{2}+1\right)$, correct. Therefore, the correct answers are $C D$.