Could she ... sorry, he (the Germans used the masculine term) have been upgraded with modern diesels or gas turbines?
My thoughts exactly. The fact that she was powered by relatively easy to swap-out diesels rather than threaded-through-the-structure steam plants would make it feasible for Argentina or Uruguay to replace them with some modern, maintainable units, and I'm sure MAN or DB would have fallen over themselves to do the design work.
A bigger problem might have been obtaining more ammunition for the main guns. The oddball 5.9" (150mm) secondary guns could have been replaced easily enough, but the 11" (280mm) guns were unique to Germany. If stocks of the ammo existed in Germany at the end of WWII, I presume they would have been confiscated by the allies: I wonder if Argentina could have done a deal for them, or maybe even for the production tooling if it still existed?
The General Belgrano was fitted with 2 x Sea Cat SAM systems and Dutch radar in 1968, so it's possible that an operational Graf Spee might have received the same systems.
Argentina already had two battleships and three cruisers with a miscellany of armament, so it wasn't like they wern't used to dealing with oddball supply chains. On the other hand, it is worth pointing out that all of those pre-war big ships, none of which had ever seen combat, let alone damage, were discarded by 1974. Only the two US cruisers lasted longer, and only one of them lasted until 1982. It therefore seems pretty unlikely that the Graf Spee would last long enough to see action in the Falklands , but it's by no means impossible, and the diesels issue might well be a factor in it's favour.
Argentinian heavy ships of reasonably modern design (a couple of WWI-era ships lasted just past the end of WWII, but only as training ships and the like) (all displacements = full load):
2 x Rivadavia-class battleships. Built in the USA, completed 1914/15, scrapped 1957. 30,000 tons, 12 x 12" guns.
2 x Veinticinco de Mayo-class cruisers. Built in Italy, completed 1931, scrapped 1960/61. 9000 tons, 6 x 7.5" guns.
1 x La Argentina-class training cruiser. Built in the UK, completed 1939, scrapped 1974. 7500 tons, 9 x 6" guns. (Note that this this ship, although perfectly capable of offensive ops, was intended and built to be a training cruiser by design.)
2 x General Belgrano-class cruisers. Built in the USA, completed 1938 (for USN), sold to Argentina 1951, one scrapped in 1977, the other sunk in 1982. 12,200 tons, 15 x 6" guns.