Capt. Robert April I believe

A whole bunch of Klingons and a handful of Federation traitors died in ST6:The Undiscovered Country so its definitely not that one.

ST1: Not sure... that Deltan chick and her squeeze didn't exactly die, but... Wasn't there a trasnporter accident in this one?
ST2: Spock, bunch of Ent crew, and a shipful of augments
ST3: Klingons, Kirk's son
ST4: Some Klingons died I think (Not sure though) <== I pick this one
ST5: Can't remember (saw it only once) but I'm sure a bunch of peons ate it
ST6: Above
ST7: Kirk, etc.
ST8: Lots of Borgified Ent crew, bunch of people on earth
ST9: Data, Romulan Senate, lots of Ent Crew, and an assload of Remans

Star Trek IV (The Voyage Home) was the only episode in which nobody died onscreen. (I'll discuss later)

The movies and their estimated body counts:

ST: TMP
Two Starfleet officers in the transporter accident. The crews of three K'Tinga class battlecruisers and the Epsilon IX station are 'digitized', along with Ilea. I don't know if this causes or counts as death or not. Decker and the Ilea probe + V'Ger join into a new lifeform, and I don't think that counts as death, unless you want to start discussing 'Tuvix (VOY)' again!
Body Count: 2 confirmed, ca. 650 likely. Also McCoy's wits.

ST II: TWoK
About twenty cadets in the Enterprise engineering spaces, + Spock. Also Khan and between 20 - 50 other "Augments", and a good dozen Regula I scientists.
Body Count: 50 - 100.

ST III: TSfS
The crew of the Grissom, David Marcus, 11 of the 'dozen officers and men' of a Klingon Bird of Prey
Body Count: About 40 - 60.

ST IV: TVH
None observed directly onscreen. This is probably the correct answer, as far as the critics were concerned. BUT..... Was every ship affected by the Whale Probe as resourceful as the crew of the Yorktown (which managed to get back in time to be retired and re-named Enterprise-A)? Or were they all rescued? Or did many of them die in space? How many people were killed in the natural disasters on Earth? Even if none were killed outright, did any in intesive care die if power to the hospitals was ever, even temporarily lost?
Body Count: ZERO. (Anywhere between 0 and tens of thousands died offscreen, though!)

ST V: TFF
Only one, Sybok, absolutely confirmed dead. BUT..... The Krazy Kat Lady doesn't seem to be getting up out of the tub/bug zapper, a few of Sybok's minions or their enemies probably get it. A couple of the Starfleet security force probably die too, unless they get medical attention immediately.
Body Count: 1 confirmed, 2 likely (Krazy Kat Lady and 'God'), about a dozen others possibly.

ST: VI TUC
Chancellor Gorkon, his aide-de-camp, probably a couple other Klingons on Kronos I, Chang + the crew of the renegade Bird of Prey, Burke and Samno (Enterprise conspirators), Colonel West, Martia, the condemned prisoner, probably several crew members on the Excelsior and Enterprise, (at least Valtane, given the events of "Flashback (VOY)", but that's not officially in the movie.)
Body Count: 8 identifiable characters, add another dozen or so for the BoP, and a good dozen Starfleeters in the battles. At least 30, probably more like 50 - 60.

ST: GEN
Kirk (only when he really dies at the end - the El-Aurian refugees were probably similarly taken into the Nexus and thus can't really be called dead), Soran, Lursa, B'Etor + the crew of the Duras' Bird or Prey, at least five on the Observatory + at least one Romulan, the crew of the Romulan outpost the Duras sisters destroyed, Renee and Robert Picard.
Body Count: 12 seen, really more like 25 - 50.

ST: FC
Hawk, Adm. Hayes, thousands of Starfleet crew members in the Typhon Sector battle, a good dozen on 21st Century Earth, Borg drones on the order of the tens of thousands, one of the Borg "queens".
Body Count: At least 10,000. Probably more like 50,000, considering all the drones offed.

ST: INS
The crew of one Son'a Battleship, many casualties on another. Ru'afo, a couple of Baku, several Enterprise officers, the misguided Starfleet admiral, several Son'a aboard the Destroyer before its surrender.
Body Count: probably between 500 and 1,500, depending on the crew of the battleships.

ST: NEM
The Romulan Senate, the crew of the Scimitar, Data, Shinzon, the Viceroy, the crew of one of the Valdore-type Warbirds, many crew of the Valdore and Enterprise, whatever "Too many" Romulans killed in the coup means.
Body Count: Between 1,500 and the millions. We just don't know how many Romulans Shinzon killed in order to get power. Nor can we be sure the crew of the Scimitar.

CONCLUSION:
Star Trek IV is really the only viable answer, but there is still a pretty good chance people did die in the episode. Otherwise, the Trek movies have been surprisingly bloody.

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Aside from Majel Barrett, the most frequently appearing actor actually seen onscreen is Micheal Dorn, as Worf!
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Janusi, I believe you are correct in your Ferengi answers. Next question?