Consider a problem 'A' which is undecidable. If 'A' is reducible to another problem 'B', then problem 'B' is undecidable.
Consider following two statements:
p: Turing machine halting problem is undecidable.
s: A given Turing Machine 'M' reaches a state 'q' on reading input 'w'.
If we can modify statement 'p' in such a way that the answer to 's' will be answer to 'p', then we can say problem 's' ( statement 's' ) is undecidable.
Also note that I will be using a term "Dead Configuration" to mean the Turing machine is in a state from which no further transitions is defined i.e, a Turing machine halts because of a dead configuration.
Modified version of 'p': A Turing machine halts only at state 'q'.
That is, whenever a Turing machine reaches a dead configuration, we will make a transition to 'q'.
Modified version pf 'p' is undecidable, which means it is undecidable that a Turing machine reaches a state 'q' on reading any input.
Hence, problem 's' is also undecidable.
On the same lines statement 's' can be modified to prove that the problem, "Turing machine will accept a string" is undecidable.
Hence, if the problem: "whether a string is accepted by a Turing machine" is undecidable then the problem of counting the number strings in a language will also be undecidable.