This is not a complete answer but if both $m$ and $n$ are at least 5, then the number of such matrices is 0.
Indeed, there must be at three lines containing at least three $1$'s each (up to relabeling the symbols). It is then easy to see that two of those lines must have two $1$'s in the same position (it is impossible to have three 3-subsets of a 5-set such that each pairwise intersection has cardinality at most one.)
If one of $m$ or $n$ is at most 2, it is pretty easy to find an explicit number. Maybe it is also possible for 3 and 4.