Johnson and his partner, His Highness Sheikh Khalid bin Hamad Al-Thani, are putting together the event.
"We want to go over there and do some match racing," Johnson said. "We haven't worked out all the details yet, but other [NHRA] teams will be invited."
The exhibition will include four Top Fuel dragsters and four Funny Cars. Johnson and Sheikh Khalid's Al-Anabi Racing team, with Larry Dixon in the Top Fuel dragster and Del Worsham in the Toyota Funny Car, will be part of the exhibition.
Johnson wasn't at liberty to reveal the other NHRA teams that would participate, but it will include some of the top teams in the sport.
Sheikh Khalid, 22, the second heir to the Qatar throne, is a drag racing enthusiast who wants to raise awareness of the sport in his home country.
"They've built a brand-new racetrack there," Johnson said. "This is their first season running events every other weekend."
The sheikh attended the NHRA season opener in Pomona, Calif., which was the first event for the Al-Anabi Racing team. Al-Anabi means "Go Team Maroon" in Arabic.
The Qatar flag is maroon and white. Doha is a city of 300,000 in a country with a population of 1.1 million.
Sheikh Khalid plans to attend two or three other NHRA events this season, but team officials can't say which ones because of security reasons for a head of state.
Worsham said the sheikh keeps close tabs on the team and called him after his victory last weekend at the O'Reilly Midwest Nationals.
"He was really excited," Worsham said. "He's a huge gearhead. You're not going to put anything past him with any fancy racing terms. He knows it all. Heck, I think he could come out and tune the car himself if he wanted to."