Forget the whole skewer thing because the bike in question does not have QR hubs and should have a solid axle.
So the question is will the bike mount to the trainer with nutted rear wheels? Probably the easiest way to find out is to try mounting the bike. There should be some kind of adjustment wheel screw that lowers the roller that the wheel touches, lower the roller and try to mount the bike on the trainer. There should be cups on the threaded axle of the trainer mount see if they will fit over the bike axle/nut assembly. The bike fit on the trainer should be solid as a rock, if not you will need to try plan B.