What is the counting complexity of random 2-SAT?
Perhaps this paper can help you:

New Worst-Case Upper Bound for #2-SAT and #3-SAT with the Number of Clauses as the Parameter by J. Zhou, M. Yin, C. Zhou (2010).

And this one that studies the structure of the set of solutions of a random 2-SAT instance: Satisfying Assignments of Random Boolean Constraint Satisfaction Problems: Clusters and Overlaps by G. Istrate (2007)