(covers information from several alternate timelines)
Asphyxia, asphyxiation, or suffocation was a condition of severely deficient supply of oxygen to the body that arose from abnormal breathing.
In 2151, Trip Tucker was concerned that a a boy was going to suffocate, when in fact his mother was weaning him off methyl oxide and onto oxygen. (ENT: "Broken Bow")
In 2154, Enterprise NX-01 discovered that the adult crew of a Xindi-Insectoid ship had suffocated when they transferred power from life support to power the on-board hatchery, protecting their young. (ENT: "Hatchery")
In 2267, James T. Kirk told Spock that without a functioning PXK pergium reactor, they had a choice of "death by asphyxiation or death by radiation poisoning". (TOS: "The Devil in the Dark")
Also that year, Khan Noonien Singh threatened the crew of the USS Enterprise with asphyxiation until they were willing to negotiate with him. (TOS: "Space Seed")
Later that year, Redjac attempted to create terror by threatening to cut off the Enterprise's oxygen and suffocate everyone. (TOS: "Wolf in the Fold")
In 2269, Lokai was believed to be suffocating due to internal atmospheric leakage aboard the stolen Da Vinci. (TOS: "Let That Be Your Last Battlefield")
In 2365, Worf bluntly told Danilo Odell that had he been standing under the USS Enterprise-D's fire-suppression system when it was activated, he'd have suffocated and died. (TNG: "Up The Long Ladder")
In 2366, Zalkonian warships possessed a weapon which could suffocate the entire crew of a starship, namely the USS Enterprise-D. John Doe was able to stop this attack. (TNG: "Transfigurations")
When Tom Paris operated a 1936 Ford truck in a USS Voyager cargo bay in 2371, the resulting high levels of smoke caused Tuvok to suggest increasing the ventilation before everyone was asphyxiated. (VOY: "The 37's")
In 2371, after the necessary destruction of the life support system, the personnel of Deep Space 9 faced suffocation within twelve hours when their oxygen supply ran out. (DS9: "Civil Defense")
In an alternate timeline in 2374, The Doctor expressed surprise that Harry Kim did not asphyxiate after twelve minutes' exposure to nebula gas. (VOY: "Year of Hell, Part II")
Also that year, Jadzia Dax told a miniaturized Miles O'Brien that he'd suffocate if he left USS Rubicon as the oxygen molecules outside were too large for his hemoglobin to assimilate. (DS9: "One Little Ship")
In 2375, Janeway ordered Paris to reroute power from the Delta Flyer's life support systems as added protection from the Borg Queen's vessel, preferring to suffocate than vaporize. (VOY: "Dark Frontier")
When the air was sucked out of Voyager's bridge by an electro-magnetic lifeform in 2376, Captain Janeway was almost asphyxiated as she attempted to reroute emergency power to environmental controls. (VOY: "The Haunting of Deck Twelve")