Can a client sue their home owners insurance if they themselves have a serious accident in the home?

It would be considered a pointless and frivolous law suit. It s not possible under color of law to be liable to ones self.

Your liability Insurance provides for coverage and defense from liabilities asserted by others. Not by ones self.

Homeowners insurance is not designed or intended to replace a major medical insurance policy.

If when you selected medical coverage when purchased your home insurance policy you could look to that portion of your policy for some coverage, although it is generally not sufficient to cover a serious injury. If you chose the economy route and did not opt for medical coverage then you'll need to rely on your health and medical insurance policy to cover accidental injuries regardless of where the injury occurred.