The sentence is correct. Not to criticize, but I don't understand the last part of the prior comment.

It depends on the situation. If the "thing" is known (specific) to both the speaker and the listener, then yes, it is correct. However, if the "thing" is unknown (non-specific) to the listener, then you should say "It's quite an exciting thing to do" However, in natural spoken english, native speakers often disregard this rule and use "the" or "an" indiscriminately (for this sentence specifically).

‘It’s quite the exciting thing to do.’ and ‘It’s quite an exciting thing to do.’ are both correct, but they have different meanings. ‘An’ suggests that it’s one of many. ‘The’ means that it is both exciting and special. It’s an informal way of saying that it’s very exciting. In both cases, you already know what you are referring to because you use ‘it’. ‘The’ adds emphasis.