Is $(\neg \exists m) (c_m < 0)$ correct way to express "There doesn't exist an $m$ that satisfies $c_m < 0$" I have seen this two successive parenthesis notation in several books but I am still not sure if it is a correct way to express such a non-existence. I also couldn't be sure  $\neg \exists$ is the correct way to express "there doesn't exist". Would love to hear ideas on how to express this in a better way, especially if the way I wrote is wrong.

Yes, this is perfectly fine and correct.
Perhaps the more natural way to put the parentheses would be
$$
\lnot (\exists m \;:\; c_m < 0).
$$
that is, the not ($\lnot$) has the outermost scope, and modifies the entire sentence, not just the $\exists$. But I do not think the above is very common in practice.
As has also been pointed out, you can simplify it by writing
$$
\forall m  \; : \; c_m \ge 0.
$$
