If two adjacent tectonic plates are moving in the same direction, is there significant interaction at the boundary?
It’s relative motion that matters. So in your case, I assume one side is moving faster than another, so you would have a right lateral or left lateral transform fault depending on which side is moving faster. It’s “significance” would depend on the amount of relative difference in motion and, as usually defined, its potential impact on man.

Note that if they are moving at the same rate there would be no fault (and therefore no boundary.) 

There has to be relative movement of the two plates towards one another for it to be “convergent.” Convergent plate boundaries are where one plate is being subducted (e.g. Japan) or pushed against (e.g. Himalayas) another. Having a combination of movements is not really uncommon, but typically of secondary importance. Good example this situation exist along the San Andreas Fault where it locally bends a bit, adding localized convergent components to what is an overall transform fault system.