Jalapeño peppers are from the Mexico, but they are an important part of Tex-Mex cuisine. Jalapeños are cited by that name in English from at least 1929.
 
The Texas legislature named the jalapeño the official state pepper in 1995. In 1997, the Texas legislature made the chiltepin the “official state native pepper.”
 
     
(Oxford English Dictionary)
jalapeño, n.
orig. and chiefly U.S.
[Mexican Sp. (chile) jalapeño Jalapa chilli: see JALAP n.] 
Also jalapeño pepper. A very hot green chilli pepper, used esp. in Mexican-style cooking.

1949 J. TRAHEY Taste of Texas 28 Homesick Hors d’Oeuvre… A bottle or can of pickled peppers (preferably jalapenas). 1964 MRS. L. B. JOHNSON White House Diary 25 Apr. (1970) 123 Hash is one of Lyndon’s favorite foods, especially with jalapeños.
     
Wikipedia
The jalapeño is a small to medium-sized chile pepper that is prized for the hot, burning sensation that it produces in the mouth when eaten. Ripe, the jalapeño can be 2-3.5 inches and either red or more commonly green. It is a cultivar of the species Capsicum annuum. The name jalapeño is pronounced IPA: [hɑləˈpeɪnjoʊ] or IPA: [hæləˈpeɪnjoʊ] in English, and IPA: [xalaˈpeɲo] in the original Spanish. It is named after the city of Xalapa, Veracruz where it was traditionally produced. 
 
Texas Legislature
H.C.R. No. 105
HOUSE CONCURRENT RESOLUTION
   
WHEREAS, The State of Texas has traditionally recognized a variety of official state symbols that embody the proud spirit and rich heritage of our state; and
 
WHEREAS, The bluebonnet, the pecan tree, and the ruby red grapefruit are examples of natural specimens that serve to exemplify the abundance and diversity of the Texas landscape; and
   
WHEREAS, In keeping with this custom, the designation of the jalapeno as the official State Pepper of Texas will provide suitable recognition for this delicious and indispensable staple of Texas cuisine; and
 
WHEREAS, Capsicums, more commonly called peppers, provided nourishment for Native Americans in central Mexico as early as 9,000 years ago and, following the voyage of Columbus to the Americas, became a popular addition to the diets of people throughout the world; and
 
WHEREAS, The most popular variety of capsicum, the jalapeno, is considered by many Lone Star citizens to be an essential ingredient in chili, our official State Dish; found in salsa and queso, the ubiquitous jalapeno is used by Texans to add zest to even the most common American dishes, and many brave souls have been known to consume the fiery foodstuff in its raw and pickled states; and
 
WHEREAS, In addition to pleasing Texas palates, jalapenos are an important agricultural crop in the Lone Star State, which leads the nation in jalapeno production and consumption; these savory peppers are also a hot commodity in the national marketplace, with demand growing at an estimated 15 to 20 percent each year; and
     
WHEREAS, Jalapenos get their distinctive sting from capsaicin, a chemical used to treat arthritis, shingles, sore muscles, and nerve disorders and believed by many researchers to be effective in the fight against cancer and heart disease; the remarkable health benefits of these small but potent peppers are compounded by the fact that they contain as much vitamin A as carrots and more than twice the vitamin C of oranges and lemons; and
 
WHEREAS, A culinary, economic, and medical blessing to the citizens of the Lone Star State, the jalapeno is widely recognized as an emblem of our state and is a distinctive reminder of our state’s unique heritage and diverse culture; now, therefore, be it
   
RESOLVED, That the 74th Legislature of the State of Texas hereby designate the jalapeno as the official State Pepper of Texas.
 
18 October 1929, San Antonio Light, pg. 8C ad:
Chiles Jalapenos (Mexican Peppers)
Small Can…28c
Large Can…48c
(...)
JOWDY’S MARKET
San Antonio’s Largest and Finest Food Market