Concerning this fact, the paper by Carmona et al. refers (on p. 28) to "Example 2 of Section 4.9.13 of Liptser and Shyriaev [19]".
A proof of this fact can also be found e.g. in Section 9.3 of this project, p. 66. The proof consists in showing that the index of $p$-variation of a fractional Brownian motion with Hurst parameter $H$ is $1/H$, whereas the index of a semimartingale must be in the set $[0, 1]\cup\{2\}$. The index is defined as the exact lower bound of the set of all real $p>0$ such that the $p$-variation of the process is finite. So, it follows that the quadratic variation of a fractional Brownian motion with Hurst parameter $H\in(0,1/2)$ is infinite. $\quad\Box$