When reading literature about the Burkholder-Davis-Gundy inequality, integrability is often glossed over.
The BDG inequality says for continuous local martingale that
$$E[[M]_t^{p/2}]\lesssim E[(\sup_{0\leq s\leq t}|M|)_t^p]\lesssim E[[M]_t^{p/2}].$$
Does this imply that if the $p$th moment of $M$ is finite and then the $p/2$th moment of the QV is finite and vice versa?
First, I just wanted to point out that the BDG inequalities also require $M_0 = 0$. Without this assumption, they would not hold as $\mathbb{E}[[M]_0^{p/2}] = 0$ for all $p > 0$.
Now, as to your question: Yes, the BDG inequalities hold without any integrability assumptions. If we set $\tau_n := \inf\{t : |M_t| \ge n\}$, then $M^{\tau_n}$ is a bounded continuous local martingale, for which the BDG inequality of course applies, so
$$E[[M^{\tau_n}]_t^{p/2}]\lesssim E[(\sup_{0\leq s\leq t}|M^{\tau_n}|)_t^p]\lesssim E[[M^{\tau_n}]_t^{p/2}].$$
The monotone convergence theorem gives that all the expectations converge (possibly to infinity) when $n \rightarrow \infty$. In particular, the inequalities still apply whether the expectations involved are finite or not.