The sentence is missing a word to make it grammatically correct, and that word would be the one to function as an adjective.
Let's look at a couple of ways to make the sentence correct and see how an adjective would fit in:
Option 1:
"Whose car are you borrowing for prom Saturday night?"
In this case, "car" is the noun, and it is not modified by an adjective.
Option 2:
"Whose red car are you borrowing for prom Saturday night?"
In this case, "red" is the adjective modifying the noun "car".
Therefore, "this" does not function as an adjective in the given sentence.