Amazon q cli autocompletion not working in cursor editor's integrated terminal

### Checks

- [x] I have searched [github.com/aws/amazon-q-developer-cli/issues](https://github.com/aws/amazon-q-developer-cli/issues?q=) and there are no duplicates of my issue
- [x] I have run `q doctor` in the affected terminal session
- [x] I have run `q restart` and replicated the issue again

### Operating system

Linux 6.8.0-60-generic - Ubuntu 24.04.2 LTS

### Expected behaviour

Amazon q autocompletion works perfectly in VS Code's integrated terminal, but not working in cursor editor's integrated terminal.

### Actual behaviour

It should autocomplete in Cursor editor's integrated terminal, too.

### Steps to reproduce

Open Cursor editor's integrated terminal and run `q doctor`.

Getting below output in `q doctor`:

```
✘ Amazon Q terminal integrations: qterm is not running in this terminal, please try restarting your terminal

Q_TERM=

✘ Doctor found errors. Please fix them and try again.

If you are not sure how to fix it, please open an issue with q issue to let us know!
```

### Environment

```yaml
[q-details]
version = "1.10.1"
hash = "cb90e381cf8391beef4b966623d913e258cbb08d"
date = "2025-05-16T13:57:39.191776Z (3d ago)"
variant = "full"

[system-info]
chip = "12th Gen Intel(R) Core(TM) i5-1235U"
total-cores = 10
memory = "15.31 GB"

[system-info.os.linux]
kernel_version = "6.8.0-60-generic"
id = "ubuntu"
name = "Ubuntu"
pretty_name = "Ubuntu 24.04.2 LTS"
version_id = "24.04"
version = "24.04.2 LTS (Noble Numbat)"

[environment]
cwd = "/home/USER/Desktop/workspace/test/test"
cli-path = "/home/USER/Desktop/workspace/test/test"
os = "Linux"
shell-path = "/usr/bin/zsh"
shell-version = "5.9"
terminal = "VSCode"
install-method = "unknown"

[env-vars]
DISPLAY = ":0"
PATH = "/home/USER/.cursor/extensions/ms-python.python-2024.12.3-linux-x64/python_files/deactivate/zsh:/home/USER/Desktop/workspace/test/test/.venv/bin:/home/USER/.local/bin:/home/USER/.pyenv/shims:/home/USER/.volta/bin:/home/USER/.cursor/extensions/ms-python.python-2024.12.3-linux-x64/python_files/deactivate/zsh:/home/USER/Desktop/workspace/test/test/.venv/bin:/home/USER/.local/bin:/home/USER/.pyenv/bin:/home/USER/.volta/bin:/opt/Cursor/usr/bin/:/opt/Cursor/usr/sbin/:/opt/Cursor/usr/games/:/opt/Cursor/bin/:/opt/Cursor/sbin/:/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/usr/local/games:/snap/bin:/snap/bin:/home/USER/.local/bin:/usr/local/go/bin:/home/USER/go/bin:/usr/local/go/bin:/usr/local/go/bin:/home/USER/go/bin:/usr/local/go/bin"
QT_IM_MODULE = "ibus"
Q_SET_PARENT_CHECK = "1"
Q_USING_ZSH_AUTOSUGGESTIONS = "1"
SHELL = "/usr/bin/zsh"
TERM = "xterm-256color"
XDG_CURRENT_DESKTOP = "Unity"
XDG_SESSION_DESKTOP = "ubuntu"
XDG_SESSION_TYPE = "wayland"
XMODIFIERS = "@im=ibus"
ZDOTDIR = "/home/USER"
```