The cuisine barbecue first appears in American English as a loanword from American Spanish barbacoa, meaning “barbecue”, around 1700. Barbacoa, reborrowed into English in 1953 as barbacoa referring specifically to Mexican cuisine, was itself a loanword, but its history remains unclear. The consensus is that it’s a borrowing from Taíno barbakoa, meaning “framework of sticks”, but the word’s popularization seems to originate in Mexico, which did not have many Taíno (or other Arawakan language) speakers. The OED instead attributes it to Haitian barbacòa (through Taíno), but there’s also speculation it’s from Mayan Baalbak’Kaab instead.
In any case, the slow cooking of leaf-wrapped meat in an oven constructed in a hole in the ground is widespread among indigenous Mesoamerican cultures. In the Yucatan, the Mayan word for the oven is píib, and the resulting cooked meat is called pibil.
Today barbecue mostly refers to regional US cuisines. The cookout event barbecue is from 1733 (George Washington writes about attending a Virginia barbicue in 1769). The grill barbecue is from 1931. Of the four major regional US traditions, Carolina barbecue goes back to at least 1760, Texas barbecue is clearly distinct by 1860, Kansas City barbecue splits off in 1908, and Memphis barbecue is well established by 1950. The clipping Bar-B-Q is from 1926, further abbreviated to BBQ by 1938. Heinz began selling barbecue sauce nationally in 1940. Barbie originates from Australian slang in 1976.