If am a paralegal and work at law firm, then overhear someone talking about how they were hurt in and accident would it be unethical to tell someone in the law firm about that person - as in recommending them as a client?

I don't think it would be unethical.  Maybe a little "ambulance chaser-esque", but it sounds like you're trying to find a solution to a problem.  Sure, it may be unsolicited, but the injured person may actually need legal representation.  They can always tell your firm no thanks.