I do not recall any plants in the mess hall or quarters in any other Trek show...
I will refresh your memory:
I do agree that the lack of plants seems fitting for the NX-01 for reasons already stated in this thread. A couple of leaves here and there couldn't have hurt, though.
They all appear to be captains' quarters or offices, rather than general crew. I'd imagine other places might be VIP quarters where you'd find such stuff. It is a drain on resources of a ship, I'd expect regulations would prevent general posession of flora and fauna (stupidity of Trouble with Tribbles aside.)
No, the one with beverly is in her quarters. There were also plants in the observation room, and there were plants in even regular crew quarters. For example there were plants in the quarters of the annoying girl that data was dating in "In Theory", and she is certainly not important or a VIP. It was hard to find photographic evidence, though, so I didn't show it. The only picture i could find on google image was this one and it's hard to tell that it's from her quarters :
I assure you there were more plants depicted in ALL of the series outside of enterprise... not just in quarters. I was just limited what I showed to quarters because he specifically was talking about quarters and mess halls. I couldn't find any plants in messhalls though, which I found surprising, because I totally thought I remembered some plants in ten forward, but that wasn't the case to my surprise.
Hot dang,
Mr. Homn really broke out the
research!
Hehe, it actually wasn't too difficult to find that stuff. Took about 15 maybe 20 minutes total. Google did all the work

.. All I did was type in variations of "Star trek plants quarters" in google image search. Voyager was definitely the hardest to find though, the best I could find was that single image with the little tiny plant in the background.
Let's also not forget Sulu was also a botanist and he was depicted with at least one plant (albiet a crappy hand sock puppet plant :P)
I was surprised that I couldn't find any plants in uhura's quarters, though, in any of the multiple episodes that depicted her quarters.. They might be there though, just hard to find on google image and I was too lazy to bust out the blu rays