The phrase is most often heard as "AT a moment's notice" or possibly "WITH a moment's notice." It DOES mean "in a very short time," but it's used in a very specific situation, when someone is going to be alerted that it's time to do something with very little warning. The "notice" part of the phrase makes it correct to be used in only this one situation. Notice means someone telling you something with time to prepare. ex. Next time you shut off the water to fix the plumbing, please give me some notice. I can't just fly to Madrid tomorrow--I have to have some notice. I wish you'd give me notice before you make a loud sound like that. Examples of "at a moment's notice": Person A: Would you like to go to the beach on Saturday afternoon? I may have to work part of the day, so I won't know what time I can go until Saturday morning. Person B: No problem! I have no plans, so I can be ready at a moment's notice. [MEANING...person B is OK with being told about the beach trip maybe 10 minutes before the meeting time. He doesn't have to prepare...he can just grab his things and be ready] Example of "with a moment's notice": Person A: Wow, you really made a nice dinner. I'm sorry I had to work late and couldn't do it as I had planned to. Person B: Well, with a moment's notice I could have made it even nicer. [MEANING...Person B is really saying "If you had given me more time to prepare by telling me you couldn't make dinner sooner, I could have shopped for better ingredients and made the meal even nicer." Person A: When are you going to take the trash outside? Person B: In a moment's notice [THIS IS COMPLETELY WRONG] Person B: In a moment. [CORRECT, A LITTLE OLD-FASHIONED SOUNDING] Person B: In a minute...or...In a second [BEST, MOST NATURAL...means "a very short time"] I hope this helps. Feel free to ask more questions. I can be ready to help at a moment's notice.

Yes but I would use 'at' a moments notice meaning without hesitation

You are correct; it means a notice (informing) in a short period of time.