I wouldn't think so. At least not quite the way you described the setup.
But if you did experiment with something of that nature, make the tubes steel instead of brass (aluminum isn't good either). You'll want a ferromagnetic material for the vibrating part. (Electric guitar pickups work by detecting changes in the magnetic field by using a standing magnet wrapped with a coil. Brass or aluminum strings/tubes won't change the magnetic field very much.) And even then, I don't think the vibration of a thick tube will be enough to register much of a signal using the pickups, even if they were steel. (But maybe it would be enough if the tubes were thin. You might want to experiment.)
[Edit: Let me correct myself. It might actually work with thick tubes (I still suggest the ferromagnetic material though). But you'll want the pickups to be under the tubes, somewhere near where the the tubes vibrate back and forth a lot -- not too near where they are constrained by the bridge. That might work.

In other words, use shorter, thick tubes, and mount them such that the middle of each tube is constrained and mounted, leaving each end to vibrate. Pickups should be underneath the tubes, somewhere near one of the ends of the tubes. You'd still only get 4 notes though.]
Also, from the way you describe things, you wouldn't be able to change the pitch of the notes. Each tube would have only a single note (and its natural harmonics). But you might be able to play it like xylophone, best case, one note per tube (although you might be able to change the tube's
tone depending on where you hit the tube, perhaps). So for a four tube instrument, you'd have a total of 4 notes. But I wouldn't bet on it working too well. Oh, and you wouldn't be able to tune it either.