I have to be on the computer for work for most of my day. Is there an eye exercise that I can do very quickly to alleviate the strain?

Adjust computer dist. Taking a "vision" break every 30 minutes by staring off at a distance helps. So does changing the distance the monitor is from you every so often. Be sure to use artificial tears regularly if you feel a burning sensation. Most importantly, change what you are focusing on regularly (screen, desk, looking out the window, etc.).
Yes. Looking off into the distance may relax the ciliary muscles, which can spasm during extensive computer use. A study in japan also indicated that warm compresses applied to the eyelids for a few minutes also reduced computer-related fatigue. Low blink rates during computer use can result in dry eyes, so application of artifical tears may also be helpful.