There are two ways to approach this question; from the aspect of watching Voyager and from the aspect of watching DS9.
In the first aspect, you should be able to jump right into Voyager.
Voyager debuted only a season after TNG ended, and the first couple of seasons of DS9 didn't introduce any major story-lines or continuity that would affect the Voyager universe. However, while you should already be aware of the Maquis from watching TNG, the Maquis were actually introduced on DS9 before TNG concluded.
Additionally, with the exception of the first episode and some passing mentions in later seasons, Voyager has no connection to the rest of the Universe by design. You're supposed to take Voyager by itself.
With all that said, Voyager is still part of canon and takes place roughly at the same time the events of DS9 take place. Especially later on in Voyager, some allusions and references might make more sense with some knowledge of what's occurred in DS9, but it's not necessary at all to understand what's going on in Voyager.
If anything, you might want to watch Star Trek: First Contact at some point in the first few seasons of Voyager. You might also want to watch Star Trek Generations so you're not lost on some background info in Star Trek: First Contact, but it's not that great a film.
With regards to the second aspect—watching DS9 unspoiled—those same passing mentions might spoil some of the plot lines of DS9 (and the Star Trek movies Generations-onward, for that matter).
Ideally, you should follow the intended chronology to avoid any chance of spoiling either series. If you wanted to follow chronology, the timeline is roughly:
- TNG Season 6 starts
- TNG episode, "Chain of Command"
- DS9 Season 1 starts halfway through TNG season 6
- TNG Season 7 and DS9 season 2 begin
- TNG Ends
- DS9 Season 3 starts
- DS9 episode, "Meridian"
- Star Trek Generations, halfway through DS9 season 3
- DS9 episode, "Past Tense"
- Star Trek Voyager Season 1 starts halfway through DS9 season 3
- DS9 season 4 and Voyager season 2 begin
- DS9 season 5 and Voyager season 3 begin
- DS9 episode, "Things Past" and Voyager episode, "Warlord"
- Star Trek: First Contact, halfway through DS9 season 5 and Voyager season 3
- DS9 season 6 and Voyager season 4 begin
- DS9 season 7 and Voyager season 5 begin
- DS9 episode, "Covenant" and Voyager episode, "Thirty Days"
- Star Trek: Insurrection, halfway through DS9 season 7 and Voyager season 5
- DS9 ends
- Voyager seasons 6 and 7
- Voyager ends
- Star Trek: Nemesis