As you have correctly guessed, this と is the quotative particle and the part before it is a quote.
A sentence that ends with を is commonly used as an expression which means "We/I/You need/want ～", "Give us ～", "Accomplish ～", etc., depending on the context.
One may say that a verb したい is omitted in this particular case, because 長生き is usually used with する:
But I don't think people bother to imagine the omitted verb in a case like this. Perhaps what you have to understand is that the noun mentioned before を is strongly desired.