Just a quick note on terminology that may help answer OP’s original question. Many people, including scientists, often use the term “adaptation” quite broadly to mean any response by an organism to a change in its environment that helps the organism. “Adaptation” in a stricter sense describes the process of evolutionary change, which OP asked specifically about. Evolutionary responses are rooted in heritable changes, and most scientists would agree that these changes will include a genetic basis (though some might think of an extended evolutionary synthesis).
There are many examples of organisms adapting (in the broad sense) to urban environments, often via changes in behavior. However, many of these examples will be examples of behavioral plasticity, or the ability of an individual organism to conduct multiple different behaviors dependent on context, sometimes (but not always) because of learning. This type of plasticity is one of the quickest ways for organisms to respond to changes in their environments, but it often isn’t evolutionary. In fact, conclusively demonstrating that these changes are evolutionary responses (ie, rooted in genetic changes) is often quite difficult, even in circumstances where we might have good reason to suspect that they are.