You can't pay the taxes on a portion of another person's property because that portion does not have a legal description, i.e., metes and bounds or lot and block. It has not been surveyed and platted off of the original property. Even if you have been mistakingly using a portion of your neighbor's property you can not file for adverse possession because that area is still part of a gazetted, legally described, property on which the owner is paying taxes.

If the owner is not paying taxes then you can apply to the taxing authority, either county or city, to have the property declared delinquent and bid on it when it comes up at a tax sale.