No, but they may contain ingredients that are rich in amino acids.
Drinks do not have any protein-rich ingredients because for purposes of convenience and absorption, the body needs to absorb amino acids through digestion rather than through your mouth. However, many drinks might contain substances that are rich in specific types of amino acids that contribute to the flavor or texture of drinks. For example milk contains casein protein which is beneficial for skeletal muscles, skin cells among others. Milk also has whey protein which can strengthen immunity against diseases. You could also find plant proteins like soy proteins in some different kinds of drink powders on the market today even though it won't show up as "protein" on a nutritional label under macronutrients.