Qt or Electron don't; they both have software renderers. The latest Godot needs a GLES 3 or Vulkan GPU. The older versions used to only need a GLES 2 GPU.
Qt and Electron output to the OS-level display APIs, where screen rendering is handled by X11, Wayland, Quartz, WDDM, etc., all of which still rely on GPUs for output.
Godot, as a game engine, bypasses the OS and uses lower-level graphics APIs like OpenGL and Vulkan. But you only need these APIs available regardless of what hardware is under them, and you can use software-based rendering with e.g. Mesa, Lavapipe, or other software renderers. With a 2D desktop application, performance should still be fine.
X11, Wayland and WDDM only use the bare minimum functionality from those APIs, so they work just fine on old DX9-level GPUs for example. Godot needs much more recent GPUs.
It's the use case that requires the bare minimum functionality. Godot might require more recent lower-level APIs than the other GUI toolkits, but if the use cases are similar, then it wouldn't require more performance out of them than the other GUI toolkits would. Using software-based implementations of those APIs would more than likely be sufficient, meaning that there's no dependence on any specific class of GPU.
