Nope, but as mentioned in other comment, I'm letting the DE handle the shortcut and it works flawless. I think I read there's some way for CopyQ to react to the shortcut on its own but it never worked for me.

Edit: Here's the other comment for context: https://lemmings.world/comment/1729453

First I have this systemd user service at ~/.config/systemd/user/copyq.service:

[Install]
WantedBy=graphical-session.target

[Service]
ExecStart=/nix/store/9qni1by8lzjccm5pc4hqpm70f3wllfqg-CopyQ-unstable-2023-04-14/bin/copyq
KillMode=process
KillSignal=SIGINT
Type=simple

[Unit]
After=graphical-session.target
Description=CopyQ, a clipboard manager
Documentation=man:copyq(5)
Requires=graphical-session.target
Wants=graphical-session.target

Replace the /nix/store/9qni1by8lzjccm5pc4hqpm70f3wllfqg-CopyQ-unstable-2023-04-14/bin/copyq with /usr/bin/copyq or wherever your copyq is installed. Then you probably need to reload the systemd daemon or something, not sure (try systemctl --user daemon-reload and then systemctl --user enable copyq.service and then systemctl --user start copyq.service - you only need to do this once).

This service needs to be running for CopyQ to work. Alternatively use any other way you're familiar with to automatically start copyq server after you log in. Trust me, there's nothing more frustrating then suddenly needing it and realizing you haven't started the server so it doesn't have the content you're looking for.

Afterwards you need to configure your desktop environment to assign a global shortcut for the command copyq menu. For example in Cinnamon it's System Settings -> Keyboard -> Shortcuts -> Add custom shortcut. Then you add the name (e.g. CopyQ), the command (copyq menu) and after pressing Add you assign the shortcut, for example Super+V. This step is different for every desktop environment, but googling "[your DE name] global shortcuts" should help.

If by any chance you have NixOS and Cinnamon, you may configure it like this:

{ config, pkgs, ... }:
{
  home-manager.users.your-username = { # installs copyq as a user package
    home.packages = with pkgs; [
      copyq
    ];
  };
  dconf.settings = { # this is for creating the global Super+V shortcut
    "org/cinnamon/desktop/keybindings" = {
      "custom-list" = ["__dummy__" "custom0"];
    };
    "org/cinnamon/desktop/keybindings/custom-keybindings/custom0" = {
      binding = ["v"];
      command = "copyq menu";
      name = "CopyQ";
    };
  };
  systemd.user.services.copyq = { # creates the systemd service
    Unit = {
      Description = "CopyQ, a clipboard manager";
      Documentation = [ "man:copyq(5)" ];
      Wants = [ "graphical-session.target" ];
      Requires = [ "graphical-session.target" ];
      After = [ "graphical-session.target" ];
    };

    Service = {
      Type = "simple";
      ExecStart = "${pkgs.copyq}/bin/copyq";
      KillMode = "process";
      KillSignal = "SIGINT";
    };

    Install = {
      WantedBy = [
        "graphical-session.target"
      ];
    };
  };
}