Monotone-2SAT and Vertex Cover The following decision problem is called k-True-Monotone-2SAT:


  Given a 2-CNF boolean formula $F$ that does not contain any negated variables
  and given a positive integer $k$, can $F$ be satisfied by setting $k$ or fewer
  variables to true?


It is NP-complete: it's easy to see a straightforward reduction from Vertex Cover to it (i.e. can we cover all the edges with $k$ or fewer nodes?).

The following decision problem is called Monotone-2SAT:


  Given a 2-CNF boolean formula $F$ that does not contain any negated variables, is it satisfiable?


As a decision problem, Monotone-2SAT is trivial. The answer is always YES: just set every variable to true.

But consider its counting version, called #Monotone-2SAT:


  Given a 2-CNF boolean formula $F$ that does not contain any negated variables, how many satisfying assignments $F$ has?


Surprisingly, #Monotone-2SAT is #P-complete.

Now here is the question. Suppose we have an oracle for #Monotone-2SAT, which returns the exact solution count of a Monotone-2SAT formula: how such solution count can be used to solve k-True-Monotone-2SAT?

I'm asking this because I do not immediately see how the solution count may give information on how many solutions have k or less literals set to true and how many don't.
The name “#Monotone-2SAT” usually refers to the problem of counting the satisfying assignments of a given monotone 2CNF formula, without a restriction on the number of variables set to true.  The stated “Monotone 2SAT” problem, or the Vertex Cover problem as is usually called, is not the decision version of #Monotone-2SAT because of the additional restriction on the number of variables set to true.  (This is certainly unfortunate.)  Therefore, Vertex Cover (or “Monotone 2SAT”) is not reducible to #Monotone-2SAT in the same way as 3SAT is reducible to #3SAT.

Note that Vertex Cover is clearly in NP and that #Monotone-2SAT is known to be #P-complete (see my answer to your previous question for the reference) and hence NP-hard.  Therefore, Vertex Cover is reducible to #Monotone-2SAT.  (Note that this does not require the fact that Vertex Cover is NP-complete.)  To construct an actual reduction, you can simply composing several reductions as always.  There may be a simpler reduction than this, but I do not expect that seeking for a simpler reduction gives much insight into either problem.