Can a 6.35 mm mic work on a computer with just the adapter 6.35 to 3.5?

Yes, you can use a microphone which has a 1/4" (6.35 mm) plug in a computer with an 1/8" socket. You would just need to use an adapter.

Now, both the plug and the socket will need to have the same number of conductors. While it might be usable to plug a standard phone plug (2 wires) into a TRS socket ("tip, ring, sleeve" -- 3 wires) or vice-versa, you likely will be missing sound from one side. In that case, there are additional adapters to convert between those types of plugs and sockets.