Police said the 40-year-old had been taken into custody for allegedly threatening the other person involved with a machete.
They were still trying to find the other person involved.
In a statement, Detective Senior Sergeant Tim Leitch said the altercation began shortly after 5pm on Wednesday outside a motor lodge across the street from the hospital.
A person was threatened with a machete by a person driving a white Ford Range. Police said the vehicle had been found and was being forensically examined.
Leitch said the threatened person then grabbed a firearm and along with another person, jumped into a grey Mitsubishi Colt and drove into the hospital car park, opposite the emergency department's entrance.
"They then drove the wrong way up the main vehicle exit stopping outside the main hospital entrance at 5.16pm where the Ford Ranger was now parked.
"A further confrontation occurred between the occupants of the two vehicles before a firearm was discharged towards the Ford Ranger.
"Both vehicles and their occupants immediately left the area."
The man in custody is due to appear in Wellington District Court on Friday charged with possession of an offensive weapon.
Meanwhile, staff at Wellington Hospital said extra security was in place.
A nurse at the hospital said the main reception was open on Thursday morning, but people were having to use a buzzer to be let into wards, and staff moving between wards needed to use swipe cards.
Leitch said police wanted to speak to anyone who may have witnessed the incident and seen the vehicles involved in the area.
"The incident occurred in a very busy public place and a number of witnesses including hospital visitors and taxis we close by at the time.
"Initial indications suggest those involved were known to each other and there is not an immediate threat to the wider public."
Leith said a scene guard was in place outside the motor lodge and there would be an increased police presence in the area.
Te Whatu Ora Capital Coast & Hutt Valley Hospital & Specialist Services group director operations Jamie Duncan told RNZ staff who witnessed the incident were being supported.
Police asked anymore with information, including photos or videos, to call 105 or make a report online.
Information could also be given anonymously by calling Crime Stoppers on 0800 555 111.
