Yes, in the sense that you are providing extra energy. No, in the sense that you will not notice much difference.
A green laser pointer has power $P$ of about 100-300 mW. Ice has heat of fusion $H= 333$ J/g: you need 333 J to melt one gram. So if all the laser pointer energy went into melting ice, it would melt $P/H=$0.0003 to 0.0009 g/s. So for a one gram ice cube you would have to keep the laser on it for about 20 minutes before it melted just due to that energy. Clearly the heating from the environment is going to be more effective.
This also assumes 100% of the energy is absorbed by the cube. Ice absorbs about 2% of light per centimetre, so the ice cube would presumably just scatter the remaining 98% of the laser energy.