Lets assume the lav top is cultured marble ( the resin stuff ). If it is natural stone, marble or granite, do not do this yourself.
But on the cultured marble, first you need to get a sink bracket set from a hardware store. It will have short expanding inserts. You drill the appropriate size hole in the countertop ( underside ), put in the insert, and then the studs which some with to secure the brackets.
Use masking tape to mark a stop on your drill bit. The countertop is probably 1/2" thick , and the insert is about 1/4". You have some wiggle room, but not a lot. If you drill through to the top, it will ruin your day!
If this was a "factory built" top with sink epoxied in place, it may be natural stone, and I do not recommend trying to drill that yourself.
Even factory applied, I do not consider epoxy alone to be a satisfactory mount for a sink. Brackets, and silicone caulk to seal are what you need.