“Quit” (and “stop”, etc.) can be used with the gerund and / or the infinitive, but the meaning is entirely different. Here’s an example with both: “Joe quit studying French to study German.” The meaning is that Joe was studying French, but he stopped — he is not studying French any more. He stopped studying French in order to study German (which he may still be studying now). Here is another example: “Mary quit (or “stopped”) running to tie her shoe.” In such sentences, either phrase can be used separately: “she quit running” or “she quit to tie her shoe”. Note that the meaning is completely different.

You can use it with infinitive. Like Phil and David explained, it has a different meaning. I would say it is infinitive of purpose. Let's say I'm talking about my job: I quit my job to focus on my health. (= in order to focus on my health)

You can’t really use ‘quit’ plus the bare infinitive. You can quit something to do something else. She quit running to let her knee injury heal. Sometimes you can omit the first something: She quit working to start a family. She quit to start a family. (‘Working’ is understood from the context)