NATO is comprised of countries that have signed on to the North Atlantic Treaty. In the treaty Article Five describes a commitment to collective defense which can be invoked by any member nation.

Article Five

The Parties agree that an armed attack against one or more of them in Europe or North America shall be considered an attack against them all and consequently they agree that, if such an armed attack occurs, each of them, in exercise of the right of individual or collective self-defence recognised by Article 51 of the Charter of the United Nations, will assist the Party or Parties so attacked by taking forthwith, individually and in concert with the other Parties, such action as it deems necessary, including the use of armed force, to restore and maintain the security of the North Atlantic area.

The treaty is clear that it only applies to armed attacks against member nations in Europe or North America. The Falklands are in the South Atlantic and therefore outside the purview of NATO's agreement to collective defense. The limited nature of Article Five was by design and specifically clarified in Article Six.

Article Six

For the purpose of Article 5, an armed attack on one or more of the Parties is deemed to include an armed attack:

• on the territory of any of the Parties in Europe or North America, on the Algerian Departments of France 2, on the territory of or on the Islands under the jurisdiction of any of the Parties in the North Atlantic area north of the Tropic of Cancer;

• on the forces, vessels, or aircraft of any of the Parties, when in or over these territories or any other area in Europe in which occupation forces of any of the Parties were stationed on the date when the Treaty entered into force or the Mediterranean Sea or the North Atlantic area north of the Tropic of Cancer.

So the short answer is that NATO did not come to the defense of the UK in the Falklands because the UK did not invoke Article Five and the UK did not invoke Article Five because is was not applicable to its conflict with Argentina.

Here is a link to the full North Atlantic Treaty on NATO's website if you want to check it out: http://www.nato.int/cps/en/natohq/official_texts_17120.htm

Simply, because Article 5 is geographically limited in its scope:

The Parties agree that an armed attack against one or more of them in Europe or North America shall be considered an attack against them all [...]

The UK did seek various kinds of assistance from its NATO allies, but often received at best equivocal support. Certainly there was no international appetite to get actively involved in a territorial war in the South Atlantic — and nor did Thatcher request military intervention by British allies.

France, a significant supplier of arms to Argentina before the war — including both the Exocet anti-ship missile and the Super Étendard aircraft the Argentine Navy used to launch them — covertly helped the British armed forces and intelligence services develop tactics to counter the Exocet threat and prevent Argentina from acquiring more, according to then-Defence Secretary John Nott. However, French technicians working for Dassault were also on the ground in Argentina to assist them in deploying Exocet, in apparent defiance of the arms embargo which France, like most of the European NATO members, introduced after the invasion. (The French have since claimed that at least one member of that team was a DGSE agent feeding intelligence back to Paris — and thence to London.)

Despite its public neutrality and diplomatic activism during April 1982, the United States also supplied covert significant materiel and intelligence support to the UK — including more than 100 state-of-the-art Sidewinder missiles and access to satellite imagery from the National Reconnaissance Office. One contingency plan would even have seen the US Navy loan the amphibious warfare ship USS Iwo Jima to the Royal Navy in the event that they lost one of their aircraft carriers.

u/Holokyn-kolokyn provided a good answer to this question. https://www.reddit.com/r/AskHistorians/comments/4gh8t4/why_did_the_uk_not_invoke_article_5_and_get_nato/d2hl8nl/

TLDR Falklands are outside the geographical area that NATO covers.

Even if the geographical area was covered, the UK would still have had to active Article 5 before NATO came to their aid.