Fast and quick mean moving with great speed. Fast is both an adjective and an adverb. Quick is an adjective and the adverb form is quickly.

We usually use quick to refer to something happening in a short time, or a shorter than expected time:

We stopped for a quick snack.

I just need a quick answer.

Fast refers to things that happen or go at high speed, e.g. a train, a person running:

She loves fast cars.

Note the common expression fast food to refer to food such as hamburgers, which are cooked and served in a very short time in restaurants:

You shouldn't eat so much fast food. It's not good for you.