Do you happen to be driving in the direction of N?

No offense, but "Do you randomly drive towards Nice?" is hilarious. I think the "happen to" construction will correspond very often to the German "zufällig" when "accidentally" is not meant. "Ich habe es zufällig fallen lassen," is, of course, "I accidentally dropped it." "Ich bin ihm zufällig in der Stadt begegnet," kann gleichgut "I accidentally ran into him in town," oder "I happened to run into him in town" sein.

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Thank you for your suggestion but my dictionary says that "happen to be" is very old-fashioned.

Any other ideas?

So, whom are you going to believe, me, an educated native speaker, or your dictionary? If you actually have a dictionary that characterizes the "happen to X" construction or the particular case of "happen to be" as old-fashioned, then you should check out some other entries in that dictionary as against other (better?) dictionaries or some native speakers. On this one it's just wrong.

old fashioned also ?

Off the top of my head, I can't think of an example where "by accident" and "accidentally" aren't interchangeable. (Excuse the double negative. I think they are always interchangeable.) Nor is either old-fashioned. And neither can be used when translating: "Fahren Sie zufällig nach N?"

If you don't like "happen to be", which is in fact perfectly ok, you could try "by any chance"

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I didn't want to insult you.
I just wanted to hear some other opinions.
Thank you!

Steve Jobs used the phrase "and I happen to have one here (the iPad)" in his presentation of the iPad. So I guess its wight.

Steve Jobs used the phrase "and I happen to have one here (the iPad)" in his presentation of the iPad. So I guess its wight.