Only insofar as they have the same source: the name of Christopher Columbus.

"America" is kind of an odd name, taken from cartographer Amerigo Vespucci. But it caught on, the way things that are kind of counterintuitive will sometimes.

This left a kind of unsatisfied urge to name it for Columbus, it seems. In 1776, a certain country was very nearly the United States of Columbia, did put their capitol in a District of Columbia, and it remained a tradition to poetically personify the country in the female figure of Columbia (songs like "Columbia the Gem of the Ocean" are about the USA).

On the other hand, there's a lot of use of Columbia (meaning the Americas) in Canada, starting with British Columbia.

Colombia (notice spelling), as Gran Colombia, and very different borders, seceded from the Spanish Empire in 1819, but split up by 1830 into New Granada, Ecuador & Venezuela. It was 1863 before New Grenada became the United States of Colombia.