‘Yunan’ in Hindi comes from the Sanskrit ‘Yona’/‘Yavana’ (or Middle Indic cognates), in turn from Old Persian ‘Yauna-‘, attested on Cuneiform inscriptions on administration and military conquests. The Achaemenid Persian Empire (6th - 4th centuries BC) had the Ionians (in Greek, Iaon-), a ‘tribe’ of Greeks who inhabited much of Western Anatolia and islands of the eastern Aegean, as their main point of contact among the Greeks. Now the word may have been transmitted before, but the earliest we have (EDIT: in Persian) is Achaemenid records from the western reaches of their empire, and for a long while the Ionians were the only Greeks ruled by the Persians. In fact the famous first ‘Persian War’ started with an Ionian revolt, a conflict which sucked in much of the rest of Greece.

The Achaemenids also conquered parts of north-western India. By the time of Sanskrit records on the Greco-Bactrian and Indo-Greek kingdoms that traced back to Alexander’s conquests, Yunan, Yona and Yavana (u and va are a common correspondence between Persian and Sanskrit). Thus the word spread from Persian to other languages of the Middle East, as well as India. Early names often stick despite much greater contact later on (despite massive contact with India, English has not switched to mainly calling it Bharat or Hindustan… but maintains the Greek-derived name ‘India’). Thus it stuck even when the Indians had contact with non-Ionian Greeks (including the ancient Macedonians).

In fact, the name ‘India’ has almost the reverse story - our earliest records of such a name (‘Hindush’ in Old Persian) are also from Achaemenid lists, and cognate with Sindh (the h <-> s correspondence between Iranian and Indic is a fairly regular one), the Sanskrit name for what we call the Indus River. Greek in turn rendered this Persian word ‘India’ and it spread from their to Latin and other European languages.

As for ‘Greece’, this is a little of a mystery. The Romans and other Italic tribes called them Graeci, and we know that Aristotle (in his Meteorologika, in Greek) specified the name of a tribe in Epirus (north-western Greece) called the Graikoi, but says they had shifted to being Hellenes, which may easily be interpreted as saying that they had been a specifically non-Hellenic tribe (other groups in that direction existed in antiquity, including Illyrians) and then were ‘Hellenised’, which might make such a transmission of the name ironic. However, we don’t know much. Greeks had massive contact with Italy from quite early on, as starting from the 8th century BC, much of Italy was taken up with Greek colonies as part of ‘Magna Graecia’, and these were founded by cities from all over Greece - but since the Graeci described by Aristotle may be among the closest to Italy, it could be seen as plausible that again they were also a point of first contact, despite much greater contact later, and this stuck. To add to the confusion, Hesiod, one of the earliest Greek writers at around 700 BC, wrote of a Graikos, a nephew of Hellen, ancestor of the Hellenes and whose sons and grandsons supposedly gave their names to the four main Greek ‘tribes’ - one called Ion, supposedly giving his name to the Ionians… though these are myths, and we do not tend to assume that the myths necessarily came first. From this perspective the Graeci might be seen to have been interpreted to be very close relatives of the Hellenes (culturally or more), who by Aristotle’s time had Hellenised.

Note that assuming the last interpretation is correct, all of these are based on regions or subgroups who were closest to the groups who have them the exonym. This is a generally common phenomenon, with another couple of extreme examples being the origins of the names ‘Asia’ and ‘Africa’ (and the common Asian term for Europeans except possibly for Greeks, ‘Farangi’, from the ‘Franks’ of the Crusader states). For the Greeks, there are other names: the Greeks of course collectively named themselves Hellenes (with a corresponding mythological progenitor Hellen) and identified further with subgroups and sub-subgroups. In Georgian, they are named ‘Berdznebi’, meaning something like ‘the wise ones’, given their reputation in philosophy and the spread of aspects of their civilisation to the Caucasus through trade (including their alphabet, which is the chief influence on the alphabets in the region), and others.

To add on, does this have any connection to the Chinese province of Yunan, or is it pure coincidence?