We solve the monomer-dimer problem on a nonbipartite lattice, a simple quartic lattice with cylindrical boundary conditions, with a single monomer residing on the boundary. Due to the nonbipartite nature of the lattice, the well-known method of solving single-monomer problems with a Temperley bijection cannot be used. In this paper, we derive the solution by mapping the problem onto one of closed-packed dimers on a related lattice. Finite-size analysis of the solution is carried out. We find from asymptotic expansions of the free energy that the central charge in the logarithmic conformal field theory assumes the value c=-2.