Yes, the verb is simply "have." We use the same verb for "dream." You can have a dream, and you can have a nightmare. We don't use "see" or "get." For "dream," it's possible to use the word "dream" itself as a verb: "Did you dream anything last night?" / "Did you have any dreams last night?" But "nightmare" is always a noun, and I've personally never heard it used with any verb other than "have."

For the conventional meaning of ‘nightmare’, Gray’s answer is correct. But using “nightmare” to mean “horrible situation” you can correctly say I’m living (in) a nightmare. I’m in a nightmare. My life is a nightmare. This feels like a nightmare.

Thanks @Nanren.

In Chinese it is the same word used for "do" or "make". Gray's answers are always so good, I am immediately drawn to try to find an exception. I had the worst nightmare last night. I experienced the worst nightmare last night. I lived through the worst nightmare last night. - ok, yes, trying too hard :) I was forced to relive ... I repeated the same nightmare ... I managed to sleep through the worst nightmare .... . As Gray said "have", "had" are the best.