It's hard to say. Many large Native cities were demolished then built over by the Spanish, so it's somewhat subjective whether to include these cities or not. St John's and Quebec are not the oldest cities in North America, they weren't founded until the 1600's. The Spanish had already established many cities by then.

If we're going on purely European-based cities, Santo Domingo in the Dominican Republic is the oldest, having been settled around 1496.

If we're including Native cities, then it becomes much trickier, since these cities were destroyed and rebuilt as 'new' Spanish cities. In this case, Ticul would seem to be the oldest, having apparently been founded around 700 BC, though to me it seems unclear whether it was continuously occupied during this time. However Cholula seems to have more concrete evidence of consistent occupation, even surviving the Spanish conquest, and it was founded around 500 BC.