Depending on how much you consider the British policy of decolonisation a facet of the Cold War, I would argue (and did in a university paper detailing the history of the Falkland dispute), that it had a part:
Until the 1960s Argentina didn't really care about the Falklands (Juan Peron was really the first Argentine leader to actually declare he wanted them "back")^1. But once the UK started dealing with Rhodesia and their UDI, that's when things actually started to get somewhat serious. Argentina was a member of the UN's Special Committee for Decolonisation, and so had influence to do something about it. This led to the 1965 UN GA Resolution 2065 (supported by 15 Latin American states). It said, in part, for both countries “"o proceed without delay with the negotiations recommended by the Special Committee on the Situation with regard to the implementation of the Declaration on the Granting of Independence to Colonial Countries and Peoples with a few to finding a peaceful solution to the problem."^2
Seeing how the UK didn't want to deal with the UN over both Rhodesia and the Falklands, it agreed, and negotiations began in 1966, and going on for a few years.^3 However, even though the UK didn't actually care much about the Falklands at this point, they weren't going to be forced out by the UN, so decided they would only follow the policy of self-determination, leaving the future of the islands in the hands of the Islanders themselves.^4
This was further exacerbated by the Thatcher government's policy to shrink the defence budget in 1981, which meant there was less military presence to defend the Islands.^5 This was made true when the HMS Endurance, an ice patrol ship stationed nearby, was pulled from there for a savings of £2 million a year.^6
From the Argentine standpoint, they felt that since the UK had just recently given up on supporting Rhodesia, abandoning 600,000 British subjects, they would not do anything to defend 1800 Islanders, especially as the British Nationality Act of 1981 revoked citizenship of about 800 of them.^7
So it depends really on how much effect you give the Cold War on British decolonisation, especially in regard to Rhodesia. I didn't touch on the Cold War myself in my paper, but I found that decolonisation and the decline of the British as a global power were major factors; some would argue that the Cold War led to both of those, or at least exacerbated them.
Notes:
George Pendle, Argentina, Third Edition (London: Oxford University Press, 1965): 165.
United Nations, General Assembly Fourth Committee, Resolution 2065: Declaration on the granting of independence to colonial countries and peoples.
Klaus Dodds, Pink Ice: Britain and the South Atlantic Empire (London: I. B. Taurus Publishers, 2002): 127.
Piers Brendon, The Decline and Fall of the British Empire 1781 – 1997 (London: Vintage Books, 2007): 628; Andrew Roberts, A History of the English-Speaking Peoples Since 1900 (London: Weidenfeld & Nelson, 2006): 526.
Brian Harrison, Finding A Role? The United Kingdom 1970 – 1990 (Oxford, United Kingdom: Oxford University Press, 2010): 40.
Kenneth Harris, Thatcher (Boston: Little, Brown and Company, 1988): 129.
David A. Welch, “Remember the Falklands? Missed Lessons of a Misunderstood War,” International Journal, Vol. 52, No. 3 (Summer, 1997): 494; Harris, Thatcher, 130.