I keep hearing from different sources that the UK was going to launch a nuclear strike on the Argentine mainland in 1982 if the conflict in the south Atlantic continued and became impossible to support the troops over there. But at the same time I hear that Trident couldn't be used without consent of the American government which wanted to avoid an escalation that could get the USSR involved since the Soviets were very concerned about Argentine grain exports to the union at the time.
FWIW in 1982 the Royal Navy had not yet deployed the Trident missile system operationally. Somewhat similar capability was provided by the Polaris system.
It would be interesting to know where you have seen this suggested, it's certainly something I have never heard before and it seems to me that even if it were true it's unlikely to be public knowledge.
Might be worth saying that the British public were not much more than lukewarm on getting into a shooting war over some place even politicians struggled to find on a map, their reaction to massive civilian casualties in support of recapturing the islands is unlikely to have been positive.
EDIT: just occurred to me I assumed you meant targetting a large Argentinian city but if the suggestion was to target the Falklands themselves that is even more politically problematic as you would kill the very people, Falkland Islanders, the whole thing was supposedly in aid of.