No, there is not really a connection here.
You have to be careful: natural-language sentences generally aren't even expressible in a given formal framework, and even when they are they don't necessarily behave as expected.
For example - assuming we're looking at classical first-order logic - the sentence "This statement is false" is never expressible in any good sense; this is due to Tarski, and is essentially an elaboration of the liar paradox.
The sentence "This statement is unprovable" is a more interesting situation. First, we can't really just say "unprovable," we need to talk about unprovability relative to a specific theory; and that theory has to be reasonably nice for provability to be talked about internally. But it gets worse: consider the theory T = PA + "PA is inconsistent." T is consistent, incomplete, and undecidable (a good exercise - remember that PA doesn't prove its own consistency and is essentially incomplete) and yet T proves "For all p, T proves p" and so in particular T proves the statement
"The statement "This statement is T-unprovable" is false."