I have just watched "50 SHADES OF GREY" and the man in the movie says "I don't do romance". You can watch him saying that, in this 30 secs Youtube video.
Now searching "do romance" in Ngram Viewer returns nothing see:
So, no formal books even use that term though I expect the term should be used s lot if it is very popular.
So, is there any difference between "I am not romantic" & "I don't do romance"? or "do romance" is just a fad phrase that was used in the movie "50 SHADES OF GREY".
Yes, they have different meanings. In the context of Fifty Shades, I don't do romance means the character is wholly uninterested in the usual trappings of attracting a mate: wining and dining, emotional intimacy, evoking tender emotions, etc. The speaker is solely interested in sexual activity. On the other hand, someone who says I'm not romantic might be interested those things but not very skilled at them.
I don't do X means I'm unwilling to engage in X. It connotes unwillingness and does not generally convey inability. See the extremely topical question "What does 'I don't do dogs' mean?" for an in-depth exploration of this phrase. This is a modern construction, so it's unsurprising that you didn't find any hits in your N-gram search.
I'm not romantic expresses a more fundamental statement about the person's nature. Romance is alien to the speaker's personality and emotions. It would be difficult, and perhaps impossible, for them to experience or express it.
There's also a subtle difference between romance and romantic. In this case, romance is an activity, while romantic is a descriptive quality. An unromantic person might well be involved in some romancing, but it would likely either be awkward for them to attempt (because romance is not part of their nature), or one sided (someone else attempts to romance them, but encounters difficulty, because they don't experience romantic emotions much or at all).
Conversely, someone who doesn't do romance may be fully capable of experiencing romantic feelings, but they would surely stifle and not express them. Likewise, they would categorically reject all attempts to romance them, whereas someone simply unromantic might warm to the idea, or at least appreciate it.
I don't do [a thing which has a verb form] is relatively recent in common usage, and (I'm pretty certain) originates in the USA.
Partly it is just a faddish colloquial form, as you suspected.
It also has some connotational differences, which I'm currently struggling to put into words. Here goes:
I am not a romantic / I am not romantic.
I don't have romance in me; I don't have romantic feelings.
I don't do romance
I may (or may not) be a romantic but, either way, I do not engage in the actions which are viewed as typical romantic gestures. Don't expect bunches of flowers, boxes of chocolates, or being asked to go on a date to the movies from me.
Note: this is very informal, and should not be used in professional environments of any kind.
When you've been somewhat obligated into doing something that isn't part of your persona, or is against your code, and also you're slightly annoyed at the obligation, then you can use "I don't do (noun)."
Here he's saying that as a boyfriend, he's expected to do some amount of romancing, flowers and other gifts and tenderness and such, and he's not that type of person, and now, he's letting her know, this isn't what you think, this isn't that kind of relationship, and it never will be. "I don't do romance."
I think the start might actually be in some movie or show that involved prostitution? Where the girl only does certain services, and if pressed into doing a different service, she'd back off and say "nuh uh, I don't do that." But it expanded a lot, and now just is kinda assertive and sassy and can be used almost anywhere.
