When I want to use the game code，I couldn’t see the Captcha.
Is there something wrong with my net or with the website？

thats not the case for a lot of people - regardless of browser setup. doesnt even work for me in galaxy (as with some others). gog is aware its broken, but all support want to do is sell you timeshare in the benefits of google captcha.

That you see the blue image, means that recaptcha isn't being blocked completely, so what Lift describes should be what should happen.
If you're using a setup which allows Google to track you, they'll know that you're a human being, and filling in the code and clicking on "Continue" will probably work. If you don't allow Google to track you, you'll get between 1 and 1 gazillion excruciatingly slow and ever more painful AI-teaching image recognition problems, and have about a 50% chance of giving up on the process before being able to redeem the code.
If you don't see anything happening after filling in the code and clicking on "continue", you'll have stumbled upon a previously unreported interaction between something causing some Google code to be blocked, and recaptcha. It might be useful to know the exact browser and browser version you're using. (Not that it'll help fix anything, but at least then we can nod sagely the next ten times this is reported.)

If you'd like to join others in telling gog to stop handing over the privacy of their users to Google in this particular way, here's the community wishlist item for that: Nip google Captcha in the bud

Every time I see someone asking for help with CAPTCHA, I assume they're a thwarted AI trying to enlist an unwitting human accomplice.

Nice try, robot!

thank you very much