Could you get cavities by not brushing your teeth for a month?

The start of: Carious lesions will occur, but your teeth will not melt away. Most likely these "cavities" will not be deep enough to see or even treat, but the decalcified enamel can further break down over time. I don't even want to discuss the bad breath you would have after a month.

Early lesions: Cavities form from acids produced by bacteria that utilize food which produce a plaque that sticks to the teeth . This process starts soon after eating or drinking. The more time the acids are in contact with tooth structure, the more the destruction. Foods such as refined sticky sugars will do much more damage in that time than fresh fruits and vegetables.

Why?: Why would you not brush for a month? If your real question is how long does decay take to develop, then it would depend on many factors. Diet, oral bacteria, salivary flow, etc all play a roll. People w a high caries index develop decay fast, while those w a low caries index, more slowly.