Can a Turing machine have infinite states?
No.  The definition of Turing machines requires that the finite-state control unit have a finite number of states.  It's not allowed to have an infinite number of states.

A machine that could have infinitely many states in its control could accept any language (unlike a Turing machine).  However such a machine could not be implemented in practice.  For these two reasons, it would not be a good model of the computational power of real computers.

In addition, once you allow an infinite-state automaton, there's no need to have any tape -- the tape doesn't add any computational power, because an infinite-state automaton can already do "everything".  For these reasons, while it would be possible to construct machines that look like Turing machines but have infinite state, there would be little point: their power would be equivalent to an infinite-state automaton on their own, i.e., every language can be accepted by such a machine.