No, you don't.
Our colors and their combination are not protected by any kind of copyright or intellectual property. You can, therefore, use them freely without any permission or any reference to Coolors
Keep in mind, though, that some specific colors could be trademarked. In fact, there are some cases where a color or a set of more colors is part of the brand identity of a company. This means that you are allowed to use that color as long as it doesn't create confusion with that brand and its products.
This is just a rule of thumb so always pay attention and ask to a legal!