qterm integration fails to initialize - Q_TERM environment variable not set

### 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

### Operating system

Linux 5.10.238-212.953.amzn2int.x86_64 - Amazon Linux 2

### Expected behaviour

## Problem
`q doctor` consistently shows "qterm is not running in this terminal" with Q_TERM= (empty), despite multiple terminal restarts.

## Environment
- Q CLI version: 1.12.1
- OS: Linux
- Shell: zsh
- Terminal: Amazon Cloud desktop environment

### Actual behaviour

## actual behavior
- `q doctor` shows: "✘ Amazon Q terminal integrations: qterm is not running in this terminal, please try restarting your terminal"
- Q_TERM environment variable remains empty
- `q init zsh post --rcfile zshrc` generates malformed/incomplete output starting mid-code
- Multiple complete terminal restarts do not resolve the issue
- Manual attempts to set Q_TERM do not persist

## Expected Behavior
After terminal restart, qterm integration should work properly with Q_TERM set, enabling autosuggestions and shell integration.

### Steps to reproduce

## Steps to Reproduce
1. Install Q CLI (version 1.12.1)
2. Run `q doctor` - shows qterm integration error
3. Restart terminal completely (attempted multiple times)
4. Run `q doctor` again - error persists
5. Check `q init zsh post --rcfile zshrc` output - shows malformed code
6. Integration file at ~/.local/share/amazon-q/shell/zshrc.post.zsh fails when Q init command is uncommented

## Additional Notes
- Q chat functionality works normally
- Core Q CLI features are functional
- Only terminal integration features are affected

Riprodurre con `q doctor` e `q init zsh post --rcfile zshrc` nell’ambiente Amazon Linux 2 e zsh riportato. Ispezionare `~/.local/share/amazon-q/shell/zshrc.post.zsh` e confrontare il suo output generato con il valore di Q_TERM dopo un nuovo riavvio del terminale. Il lavoro è completato quando l’integrazione generata è completa, persiste quando viene abilitata e `q doctor` segnala qterm come in esecuzione.

Scritto dal modello di indicizzazione a partire dal testo della issue.