|NO, do not do that. I doubt the surge protector is large enough to handle an AC unit.|
At any rate you really don't even want the pull power for your computer from the same outlet as the AC unit.
If possible it would be best to isolate the electronics by connecting them to a different household circuit than the AC unit.
That may not be possible but if lights dim when the AC unit kicks on, you should try to avoid using that circuit for your electronics.
A line conditioner may be in order if the above conditions exist. Keep in mind that all surge protectors have a rating on them and are not all of equal quality.