According to the staff at my hotel, the answer is "technically no, practically yes"
Specifically, within the Dubai metropolitan area, but just over the border. In the Sharjah "Heart of Sharjah" / Sharjah Art and Heritage area is the large Sharjah Art museum, which houses a collection of contemporary and modern Arabic and Islamic art. Just round the corner is the Sharjah Museum of Islamic Civilization, which in the upstairs galleries features arts and artifacts from the 1st century AH / 7th century AD through to the 14th century AH / 20th century AD.
It looks like you can get a bus from Dubai to just along the Corniche from these museums, otherwise it's a fairly quick taxi ride away. (15 minutes and 60 dhs from my hotel near Dubai airport)
The only thing to be aware of is that not many Dubai taxis go there, so you'll save time if you have a map with you to show the driver. They seem to know the Sharjah Museum of Islamic Civilization a little better than the Art Museum (possibly because it's right by the water), but even then may be a bit hazy on the locations. There are a few maps up to direct you around the area, or there's an information desk near the water in the centre of the heritage area, or pop into the first museum you come across and they can show you the way to any other one!
For the Sharjah Art museum, the building you want is big, but a little tucked away down a side street, which is why you might need a map. You're looking for this:
For the Sharjah Museum of Islamic Civilization, you're looking for this on the Corniche by the water: