Can you have inhaled foreign body without having symptoms of choking?

Yes - smaller FB: Smaller foreign bodies can sometimes be aspirated (especially by children) with minimal symptoms at the time of aspiration - like a brief coughing fit while eating. I've personally seen cases in children with aspiration of peanut fragments, watermelon seeds etc. Which were causing chronic cough & wheezing but no choking episode was recollected by parents. These were diagnosed on bronchoscopy.

Under anesthesia: People inhale tiny particles every day that never trigger the cough reflex. Things like the particulate of air pollution or smoke will often not trigger much of an effect.The lung eventually cleans these up over time. General anesthesia will remove the cough reflex while under, but you will not inhale a foreign body of any size without triggering cough.