With Blender 2.8+ you can use the Auto Merge Tool to insert a vertex between intersecting edges.
Just enable the option Auto Merge and Split Edges & Faces in the Tool Shelf, then select a vertex and move it. Upon placing it down the tool will create vertices between the connected edges and the edges intersecting them.
If you don't want to move any vertices just select one,the intersecting edges or their vertices and press G to move and confirm with left click without altering the position.Enter