Now, it is working.
From Option, I selected "Headset".
(And made a small change in Recording devices setting).
Yes, you need an adaptor.
These are the three main types of connector for mic/speakers/headset
Image from CableChick - Understanding TRRS & Audio Jacks
These are generally referred to by the number of 'stripes'.
Left is Tip, Sleeve - TS - usually used for a mono mic
Centre is Tip, Ring, Sleeve - TRS - used for regular stereo headphones
Right is Tip, Ring, Ring, Sleeve - TRRS - used for a combination headset/mic.
Your laptop uses a single TRRS socket - headphones left/right, mic & ground.
Your desktop has two sockets
TRS for speakers/headphones - left/right, ground
TS for mic - mono, ground [These are sometimes wired with another TRS, but the functionality is the same to the end user]
When you plug your TRRS headset into the desktop's mic input, it's going to be fairly random as to which of the rings is going to connect - in your case, i'ts one of the speakers.
Just search for "headset splitter" - input will be TRRS, output often both TRS, but this should work.
Best Answer
In Skype, if you go to "Call" on the top menu, then click "Audio Settings," you will come to an options menu as seen below. You could select different speakers, such as your integrated laptop speakers. You could also turn the volume all the way down so that you cannot hear the speakers. There are other options if you hit the "Show advanced options" button that may help you too. These options can allow you to set up a priority of which speakers to use if they are connected.
-Hope this helps.