The apricot or Prunus armeniaca, is named after Armenia, and has been cultivated in that area for a very long time. However, it was probably first domesticated in central Asia.
The Prunus armeniaca or common apricot is the most commonly cultivated apricot species. Other species are available that are native to China, Siberia and Europe [source].
Like the fruit, the word apricot has undergone quite a journey to arrive in English. It started in Latin as persica praecocia (“early ripening peach”), then moved into Greek as πραικόκιον (praikókion – apricot). That became βερικοκκίᾱ (berikokkíā – apricot tree) in Byzantine Greek, which was borrowed into Arabic as اَلْبَرْقُوق (al-barqūq, – plum), then into Andalusian Arabic as الْبَرْقُوق (al-barqūq – the plums). It jumped into Spanish as albaricoque (apricot), and into Catalan as a(l)bercoc (apricot). It was then borrowed into Middle French and became aubercot and later abricot.
Finally it arrived in English in the 1550s as abrecock, which eventually became apricot [source].
A word from the same roots is precocious, which originally referred to flowers or fruit that developed or ripened before the usual time, and later came to refer to people and other things [source].
Other words from the same roots include biscuit, charcuterie, concoct, cook, cusine, kitchen and kiln in English, kepti (to bake, roast) in Lithuanian, and poeth (hot, roast, cooked) in Welsh.
The ultimate root of these words (or at least parts of them) is the Proto-Indo-European word *pekw- (to cook, ripen) [source].