You are confused on several accounts. The language of a Turing machine $T$ consists of all inputs $x$ such that when $T$ runs on $x$, it terminates in an accepting state. (There are several other equivalent definitions.) In particular, the language of a Turing machine can be empty no matter what the alphabet is. All it has to do is never terminate in an accepting state.
You are confusing the empty alphabet (which doesn't exist - we always require the alphabet to be non-empty), the empty string (which is the unique string of length 0), and the empty language (which contains no strings). In particular, the language consisting of the empty string isn't empty, since it contains the empty string.
The emptiness problem asks to decide, given a Turing machine $T$, whether the language of $T$ is empty or not. All you are given is the Turing machine itself. You aren't given "hints" like in your example. You can prove that the emptiness problem is undecidable. Indeed, it is even undecidable whether a given Turing machine accepts the empty string (we don't care about any other input).