Plugging a 65W computer power plug into a 45W step down transformer
Short answer: No.

If the computer requires 65 W and your transformer can only handle 45 W, then various bad things could happen.  The transformer could saturate or overheat or have the output voltage sag.  Too low a voltage is not good for the computer either.  It should not be damaged if properly designed, but might not turn on at brownout voltage or abruptly shut down when it needs the extra power.

However, check whether your computer really is 110 V only.  Most modern machines use "universal" input power supplies that can usually run from 90-260 V AC at 50-60Hz.  If this is the case, all you need is a localized line cord.