At the dentist they used a water pick to clean my teeth and water was spraying over my nose occasionally. can you get naegleria fowleri from this?

Cavitron usage: The equipment you describe sounds like a cavitron or prophyjet, which is used as an adjunct to clean teeth. The water in the unit is self contained and is normally distilled water, that is changed either daily or more than once per day. I have never experienced any patient having a bacterial infection from the device. Usually, this is done as a supplement to hand scaling.

Possible, not likely: Naegleria Fowleri infections may occur when contaminated water from other sources (tap water, dental unit, swimming pool, etc) enters the nose.

Unlikely: Especially since there have been zero case-reports of Primary Amebic Meningoencephalitis caused by Naegleria fowleri in the state of New York between 1962 to 2016. REF: https://www.cdc.gov/parasites/naegleria/state-map.html

To my knowledge, there is no reported case ever of a dental office procedure causing an amoebic infection.