Can I ground an extension cord using a two-prong adapter?

No. A two-prong adapter (cheater plug) is not designed to add grounding to an extension cord. The adapter’s metal tab must be screwed into a grounded outlet’s cover plate screw, but adding an extension cord breaks the ground path entirely. Use a three-prong extension cord with a properly grounded three-prong outlet instead. If you only have two-prong outlets, have an electrician update them or use a GFCI adapter—don’t hack it with a cheater plug.