On my DSLR, there is a setting to adjust the LCD screen's brightness level. If I were to turn this setting up (make the LCD brighter), does this affect the quality of the picture? I would assume that something looks nice at a "brighter level", is going to look darker than I expected once I transfer it to the computer. Is there an appropriate method to compensate for modifying the LCD brightness? (maybe turn the exposure down a stop?)
LCD brightness affects your perception of the photo, not the photo.
At worst, a poorly adjusted LCD will cause you to make bad settings decisions. Make your decisions based on data in the histogram to avoid such errors.
The goal when adjusting brightness is that white areas emit as much light as a white object would reflect. This means the LCD should be dimmer in low light, and brighter in bright light.
Proper adjustment is difficult to maintain. It's impossible to perfect without hardware. Just stick to the histogram.
While LCD brightness setting is not directly linked to exposure, it still may have some effect on resulting photos.
Brighter display uses more current, so the number of photos that can be taken before depleting the battery will go slightly down with a brighter LCD setting.
By using a bright display in dim environment, you will effectively dazzle yourself by looking at it and this will negatively affect your ability to compose (and manually focus) a shot for a short period of time after that.
A display too dim may let some problems (e.g. wrong focus, distracting elements, blur) on photo stay unnoticed; this is especially a problem when using Live View to compose.
The LCD brightness is in no way related to the photographs taken by your camera.
It's usually just a way of saving some battery power as well as being able to bring down the brightness in dark locations.
However, it may cause you to think your photo is unexposed when it potentially isn't, if used on a low brightness setting. In this case, it's better to check the histogram. Mine is at the default (halfway) brightness.