It's possible that Chrome OS does not support this.
The Godot Linux executable files are not actually installed on Linux. You download the file and execute it.
On my Linux distribution, which is Debian-based, I download a file such as Godot_v4.3-dev5_linux.x86_64.zip, extract the contained file Godot_v4.3-dev5_linux.x86_64, and execute it by double-clicking on it in the file manager, or execute it from the command line:
$ cd DIRECTORY_CONTAINING_THE_FILE
$ ./Godot_v4.3-dev5_linux.x86_64 -p
Does that work?
If not, you may need to ask for help in a Chrome OS support place. If you do that, provide this information so that they'll know what kind of file you're trying to execute:
$ file Godot_v4.3-dev5_linux.x86_64
Godot_v4.3-dev5_linux.x86_64: ELF 64-bit LSB executable, x86-64, version 1 (SYSV), dynamically linked, interpreter /lib64/ld-linux-x86-64.so.2, for GNU/Linux 5.15.0, stripped
Alternatively, Godot is available through various package managers, but not necessarily a current version.
apt: https://packages.ubuntu.com/search?keywords=godot&searchon=names&suite=all§ion=all
flatpak: https://flathub.org/apps/org.godotengine.Godot
snap: https://snapcraft.io/godot4
But if the executable file cannot be run using the above instructions, I don't know if these would work either.