You can use FL Studio with a Gameboy soundfont available from here. If you're going for authenticity, you'll need to keep the GameBoy limits in mind, especially with only allowing one note at a time and only having 2 square channels, a noise channel, and a wave table channel playing at once.
If you want genuine authenticity and aren't unfamiliar/afraid of trackers, you can try a GameBoy tracker. There's a list available here. Since they run on actual hardware (or an emulator), the sound would be 100% authentic. I only tried Fat Ass which is a ROM. I couldn't figure it out as I'm not very well versed in trackers outside of FamiTracker, but I managed to load an example song it provided and sure enough it was authentic.
You could also try this GameBoy tracker for PC, but I couldn't figure it out for the life of me, however I'm sure some dedication would fix that.
I would personally recommend just using a soundfont with FL Studio/DAW of your choice as the process is much simpler.