HERE
you'll find the specification of a
Turing machine
that will endlessly calculate all numbers of the Fibonacci sequence.
The current number is printed left-most on the tape.
The alphabet is {*,B,#,0,1}, the set of states is {-4,...,32},
the start state either -4 (in case you start
with an empty tape) or 0 (in case you start with the first 2 numbers typed
as 1#1 already printed on tape). There is no accepting state since the
calculation is supposed to go on forever and there is no rejecting
state either.
The numbers are represented in binary, so for the calculation there is
also a general binary adder implemented here.
The rest is mainly moving and deleting.

As simulator you could for example use
this one
that was on the DVD "Coding for Fun" by Gottfried Wolmeringer.
It was stated in the book, that this is free software.