The painting is a reproduction. The original was sold at a convention before TNG aired.
I found this out by asking Rick Sternbach directly, via his professional web site.
In my email to him:
I'm writing to you because I am a member of "Scifi StackExchange", a question and answer forum on sci-fi works. On the site, there have been a couple of recent questions regarding a painting that appears on one of the set walls of the Enterprise-D (Shelby's guest quarters in "Best of Both Worlds"). This painting is the cover that you made for "The Anvil of Jove".
We're wondering: was the painting in the guest quarters the original painting? Also, what was the fate of that particular painting? Was it auctioned off?
Thanks for your time. Also, I hope it's okay if I post the answer to the forum.
His response:
Heh. We never used originals on set, of course (with the exception of the Enterprise in Picard’s ready room, and that painting was locked away after filming), but large prints made from color transparencies and negs. The original for Anvil of Jove was sold at some SF convention ages before TNG. I just gave our set decorator a pile of images and he picked ones he liked for the sets.
Thanks for the interest, though. Were a lot of fun years working on those shows.
All the best,
Rick
Summary:
- Original painting: sold off at some sci-fi convention (exact whereabouts unknown)
- Reproduction seen in TNG: whereabouts unknown
Since not a lot of artwork used in the Enterprise quarters have gone to auction (as per my research on the Star Trek Prop Authority and other similar groups), I assume that it is either in the hands of the set decorator who made the reproduction, or was kept by some other TNG personnel, or is under lock and key for future auctions or Star Trek conventions, or was pulped. The last option is the least likely, but still possible. The second-to-last option is not likely either, considering that Rick made a point of mentioning that the Ready Room painting is the kind of decoration worthy of being locked away.
So my best guess is that the reproduction seen in the guest quarters is in the hands of some (unknown) member of the TNG design crew.