The original Bank Quay station (see
Disused Stations) was a bit to the north of the present one, near Bank Quay Bridge.
This NLS large scale plan is one of a set from around 1850, showing the first station. On the next one to the west you can see that Bank Quay is the name of an area of the town. So Bank Quay Wharf might be named after that, like all the factories, and the school, which also have Bank Quay in their name.
Alternatively, there is a Bank Hall, next to a Bank House, shown on the same map sheet as the station and a little further away from the river. Bank Hall dates from before 1800. But both of those might have been built for "new money" mill owners and named for Bank Quay Road.
Either way, by the time of the first station (1837) a lot of rapid development and redevelopment had already happened, so you can't really see what came first.