In this paper, we obtain a new class of $(n+1)$-dimensional magnetic brane solutions of quasi-topological gravity in the presence of exponential and logarithmic nonlinear electrodynamics by a spinning magnetic branes with one or more rotation parameters. For the spinning brane, the brane has a net electric charge, when one or more rotation are non zero, and also this electric charge is proportional to the magnitude of the rotation parameter. However, when all the rotation parameters are zero (static brane), the electric field vanishes and the brane has no net electric charge. In the class of solutions, we have a spacetime with an angular magnetic field. These solutions are horizonless and have no curvature, but there is a conic singularity with a deficit angle. In addition, these two forms of nonlinear electrodynamics theory, have the same behaviors for the obtained solutions. Finally, we use the counterterm method and compute conserved quantities of these spacetimes.