I’m sure LH is right: the phrase began life as a verb phrase and became nouned (with the noun version referring to an instance of meeting-cute. But the precise details of the history of the term aren’t clear.
Here is a link to a Wikipedia article on meet-cute:
The article itself use meet-cute as a noun phrase, and it includes a quote of Roger Ebert using it as a noun phrase, and a quote from the character Arthur, from the rom-com “The Holiday”, in which it is also used as a noun phrase (incidentally, I became acquainted with the phrase by watching that film).
But it also includes a side panel with a quote from a 1964 film review in which it is clearly used as a verb phrase, “to ‘meet cute’”.
Interestingly, and annoyingly, the wiki mentions an interview of Billy Wilder in which he uses the term in reference to a movie he worked on that came out in 1938, but it doesn’t include a link to the interview. But I managed to google it, and here it is:
Wilder quote himself as saying, “I have a meet-cute for your story,” clearly using it as a noun phrase. Yet the interview is from 1996, and I wouldn’t expect Wilder to remember the exact wording he had used roughly 60 years ago so this is hardly reliable evidence of how the term was actually used in the mid to late 1930s.
If I had to wager money on it I would bet that the phrase began life as a verb phrase. Whether it became nouned relatively early in its history (by the 1930s) or much more recently is harder to say (and would require far more research than I have done or have time to do at the moment).
Off topic, but an interesting side note: the character Arthur, in the movie “The Holiday” gives, as an example of a meet-cute,’a scene in a movie where man and a woman who want to buy pajamas from a department store, but one only wants a pajama top and the other only wants a pajama bottom. It turns out that this is the meet-cute in the very film that Wilder mentions in his 1996 interview, and Wilder discusses it in that interview in almost precisely the same way Arthur refers to it in the movie.