Hello, I have the same problem and I see I did the same things as you talked here.
My WP is 4.3 atm and qTranslate X is up-to-date.
But what I noticed, it works OK on Firefox, but Chrome and IE have this problem.
If I type in the URL manually in the address bar for my default language it works, but via link…nope
Tried both my own link and the default language switcher…same issue
Looked at the .htaccess for anything, found nothing…