Can opponent trash a copper on my Governor if there is no one value card in play?
Yes, they can.

You always follow all instructions in order, only skipping things that are impossible. If you choose this option, your opponent has 2 instructions to follow:


They may trash a card from their hand
They gain a card costing exactly $1 more


They follow the first instruction, and then the second one is impossible, so it is ignored.

Note that this is common to use as a benefit for yourself, with cards like Remake or Upgrade; where you can trash Coppers away.

Yes, this does mean that this option will help your opponent.. however, it would help in general anyway; that's the whole concept behind Governor. While they do have the ability to trash a Copper and gain nothing in return, they would still be better off trashing an Estate and gaining a $3 cost card. Or later in the game, trashing a $4 and getting a Duchy.

If they happen to have nothing in their hand that is worth trashing, then you lucked out and they don't get a benefit. But in general, any of the three options on Governor are designed to help your opponents a little, and help yourself even more.