Quark would talk a big game about using it or selling it (or both) but then just lock it up or something and pretend it never existed. Despite all appearances, he does have a conscience.
Quark's Death Note would be one good rounded episode of him creating a market of paid assasinations – before a real pro assassin who is out of contracts due to this would take it and, threating to put Quark in the list, would steal it for themselves and disappear.
Garak just writes, "See included item labeled 'Exhibit 1' admitted to the record." along with a 1200 page document he's spent several decades amassing.
Death, even at this time, is a disturbingly usual occurence. If I'd mourn every poor soul dying in the universe, I'd have no time running that shop. How can I afford to even shed a tear? That's, my friend, is probably a little lesson to you. If something is as inescapeable as death, you can't say it's untimely or unpredictable. That's just terms of our contracts, and let's treasure our time for it runs out as we speak.
You can probably put a delay according to their birthdate to make an abortion. I'd be only troubled how a chain of deaths can cause a butterfly effect on the situation.
Also, if DN's effect is imminent, could it also prolong life by setting it decades after?
Series aren't liked for being smart, but it's interesting to explore these terms.
The rules of the death notes generally have time limits in the tens of days. They can only control+kill a person for 23 days, and the pre-filled death methods (such as how Light killed Raye Pember) are said in supplementary text to only work for 19 days.
I imagine that first clause would prevent any significant life altering shenanigans.
I can see O'Brien thinking about using the Death Note, followed by Keiko yelling "Miles!" at him and hence resigning himself to beam it into space.
I doubt any of the DS9 characters would end up with a god complex like Light does, but I agree with the other comments that Garak would probably get the most mileage out of it and come up with some very inventive ways of dispatching people. Sisko would probably work out that Garak had the Death Note, but never be able to prove it nor that he'd used it.
Garak absolutely has the skill and resources to do it without the death note(see the whole romulan senator thing). Moreover, his dad would see it as cheating and probably disapprove.
I think he'd tease Dukat that he has it, pretend to destroy it in front of Dukat, then in the Dominion invasion he'd kill "The Great Link."
Legit opinion. I find the second half being like that, and I've seen many hold the same opinion. If you care, I tried to remember it and highlight some points.
Solid art direction and big animation budget (was it Madhouse studio?);
Like many media, they are selling two characters-stereotypes and their dynamic - hard-disciplined OCD narcissist and his complete opposite (that's why after one dies show loses my interest), both looked different to what average movies do;
Writers took a couple of twists about how Light creatively secures himself and kills people with notes and they are fun to see as much as Sherlock's explanations of his logic solving a crime;
But what I want to single out: it was made for teenagers and young adults. Confusion about their lives, wish for some superpower or direction, being able to overturn the frustration of their first boring work. With a shade of phylosophical themes that make them feel like it makes them clever, with guessing what if they end up in the same position, with bleeding edginess of deaths, god complex, acts that can be rightfully called terrorism (plus it's geeky and cool if you watched it when anime wasn't mainstream at your place - it's a kind of a community, and for some - their personality trait). If you don't consume a lot of media and DN became your first anime somewhere between 12 and 22, you'd probably like it.
And it usually kills it when you leave this age group, don't like edginess, fog of intelectualism, not only ill but just immature characters, everything turning around them and don't feel like watching episodes just to see these characters' tricks and quirks. And what can come from psychotyping (yeah, this geeky horoscope) there's no emotional information coded into the series, just pure 'we see these smart men outsmarting each other' and rotating the rules of Death Note like a rubik's cube that is problematic to many people - I'm the opposite, I don't find relationship drama even IRL interesting, but I understand the opposite can be as empty on the other end.
So, as all media, it's a matter of taste and background.