Complement pair of a lattice will have both LUB (Least Upper Bound), and GLB (Greatest Lower Bound). LUB and GLB of a lattice are unique, and in this case, they are $12$ and $1$ respectively.
A complement pair of lattices will have $12, 1$ as LUB and GLB respectively.
But here we see pair $(4,6)$ which has LUB $12$ and GLB $2$.
$(2,3)$ have LUB $6$ and GLB $1$.
$(3,4)$ has LUB $1$ and GLB $12$.
Other pairs need not be compared since they are directly joined with a line.
Therefore, I think that the given lattice has $1$ complement pair.