"Birmingham" is not a style of painting, but a city in Alabama. The phrase "Birmingham painting" is a reference to the song "Paint Me a Birmingham," written by Buck Moore and Gary Duffey. In the song, a man asks a painter at the ocean to paint him a picture, which he calls a "Birmingham."
In the picture the man describes, it is springtime, and there is a house with a porch on the edge of town where a woman wearing a white cotton dress is sitting in a swing in the front yard. The man expresses his longing to still be in a relationship with the woman and asks the painter to depict him holding the woman in his arms.
The song "Paint Me a Birmingham" has been performed by country singers Ken Mellons and Tracy Lawrence. The two artists released their versions of the song within a week of each other, but Lawrence's version was more popular and reached the number four position on the Billboard's U.S. Hot Country Songs chart in 2004. The comedic country singer Cledus T. Judd also had a hit with his parody of the song, titled "Bake Me a Country Ham," which reached number 58 on the Billboard charts in 2005.