This might work but it will be very messy and probably not give you the best results. First of all, your top mount sink was most likely caulked around the edge, between the sink lip and the granite surface. When you lift the sink, you will see a caulking stain that will most likely be impossible to remove because it has been there for a long time.
Secondly, a fabricator will have to re-cut and polish the opening to the fit the new undermount sink. A topmount sink cut-out is usually larger than an undermount polished cutout so we doubt you will find a large enough undermount sink that will allow you to do what you want.
If you want to pursue this further, start by measuring the already cut opening in the existing countertop. You might have to get under the sink to measure how and where it was cut. Then look for an undermount sink that is bigger than the opening and still fits within your cabinet size. If you pass these two hurdles, go ahead and pull the sink up to see how bad the caulking stain is.