Contact the city government of Birmingham. The city has a building department that must approve building requests, including water wells. Keep in mind that the city offers municipal water and sewer services, so your request for a water well will be scrutinized by city building officials. There may be permit fees involved at this level.
A caveat is that in a city like Birmingham, which is heavily developed with few areas for new builds and has an extensive water and sewer system, the city may not even consider your water well application.
Contact the Oakland County Health Department Environmental Health Division if city officials approve of your request. In Michigan, the county health departments deal with aspects of wells that include whether or not the specified area is safe for drinking water and tests the well for bacteria or contaminants. This is true even in a city like Birmingham, which has its own municipal government. City building officials, if they approve of your well request, will advise you on contacting Oakland County for the proper health-related permits.
If you get the go-ahead from the city, Oakland County officials will issue the permits for the construction of a new well and also well inspections. There are fees for new wells and replacement wells. The county does allow some of the permits to be applied for online. Be aware that there are "enhanced" fees for these services being offered more conveniently.
Make sure you choose the best contractor for the job. Drilling a new water well is not an area where you want to skimp on costs. This is an expensive proposition and if not done correctly will leave you with a dry well or contaminated well. County officials will be looking at your well to make sure it is safe to be used for drinking water.