I need to know whether one could say he "got a dream of" something instead of saying that he "dreamt of" something. Would the former usage be grammatically correct?
"I have a dream..."
- Martin Luther King
One normally says "I have a dream" or "I had a dream".
I had a dream. You were in it, wearing a horse costume.
This is grammatically sound, as is "dreamt about".
I dreamt about you again last night. You were still wearing the horse costume.
Using "dreamt of" is a fairly old-fashioned expression, and usually has connotations of romance if you use it with people.
I dreamt of you last night. Did you dream of me?
I have never heard anyone use the term "got a dream".
It might be grammatical, but the point is that no one would say it other than in some literary context or other, and even then it would be unlikely.
As noted by others, it would not be grammatically wrong. However:
Getting implies giving or distribution—if you got a dream, who gave it to you? (And if you know, it might be better to state that he gave it to you, rather than simply that you got it.) Where did you get it?
Got further implies current possession (whether by definition or if that's simply the usage I'm familiar with, I don't know). If you say that you got something, that implies you still have it—I got a new car; it's in the driveway. This isn't true, however, of dreams, even recurring ones; if you are not currently dreaming it, you don't have it, do you?
As noted elsewhere, it is more common to say that you had a dream.
"I got a dream of you" definitely sounds wrong. You could say "I had a dream about you," though.