Not very related. Boredom and depression are very different concepts. If you truly have depression, then there is something more going on than just boredom. Anyway, solving boredom on the job, depends on the nature of the job, and the individual. This is worth talking with a professional, as well as coworkers and others familiar with your industry and position.
Demoralized? You are more likely to get demoralized than depressed and in today's economy one needs to thankful of having a job. Recently published study showed that a group of people who wrote 20 things they are thankful for that day for 6 weeks improved mood compared to who did no do this exercise. Try doing that in addition to regular exercise preferably out door to prevent depression.
Increase challenge. Boredom often stems from being under-challenged. Try to find way to increase the challenges in your day. Come up with your own ideas for making your job more interesting. Maybe additional education will help as well. You can try applying what you learn in the work setting. You might also consider changing jobs. A boring job is not for you.