Is there not a part of the FAQ to do with this?
In the website help I found this:
"10. How do I cancel my GOG.com account?
We have no idea why you'd want to do that, but if you will ever want to cancel your GOG.com user account, please contact us via the "General / Website questions" contact form. After verifying you are the owner of the account in question and confirming that you want your account to be permanently deleted, our support rep will delete the account for you. Please remember that without an account, you won't be able to re-download your purchased games as well as any additional content available, including updates and patches. Also, it won't be possible to “recreate” the deleted account later, if you change your mind."
Or perhaps you could email GOG directly? (there is a "contact us" link at the bottom of the page).
Contact Us Link.