Not really. The film's writer Peter Morgan admitted he made that part up.
The way Niki met his wife was not in the way I showed it, but I said [to them], “Does it truthfully represent the nature of your relationships, the atmosphere and the vibes between you?” And they both signed off on it.
It helped that Morgan's wife is Austrian and actually knew Lauda in person.
Niki Lauda himself, however, said in an interview that the story is close enough:
In Rush, we see you meeting your first wife Marlene after you picked her up while she was hitchhiking. Is that a true story?
Actually I met her at a party, but I did drive her somewhere soon after – and she did not recognise who I was. She thought I was a tennis player.
The movie also shows you picking up hitchhikers and scaring them half to death when you drive at extreme high speeds?
That part is true [smiles]!