wipers

No the wipers are not connected to the transmission

there is such a thing as a "wiper transmission". it is the linkage assembly that is connected to your wiper motor.if that fails or is out of adjustment your wipers may not stay synchronized or you may here the motor running but no movement of the wiper arms.if you turned your wiper switch on when the wipers were frozen in place by ice or excessive snow on them ,that may have caused the problem.