Moved here as this isn't specific to Linux Mint.
Qt (pronounced "cute") and Gtk are application frameworks. Basically libraries that developers can use to build graphical applications. There are other application frameworks for Linux, like EFL, but probably Qt and GTK are the most used. There are also applications that don't use an application framework, and have their own code for building a graphical application (like IIIRC right now Chromium and Google Chrome).
Regardless of what desktop environment you use (e.g., Cinnamon, KDE, MATE, Xfce) you can use any graphical applications--independent of what application framework they use. Historically, most of the applications you use on Cinnamon, MATE, and Xfce are Gtk based while most applications you use on KDE are Qt based. But VLC for example is a Qt application, and it works just fine on Cinnamon, MATE, and Xfce. The Mint tools are all Gtk applications, but work fine on KDE. No biggy.
Linux users gossip and speculate too much, and they do love them their various wars and controversies
(I'm guilty of it also.) If it isn't a distro war (Ubuntu vs Debian vs Mint), it's a desktop environment war (classically, Gnome vs KDE), or a application framework war (right here! Qt vs Gtk), or a display server war (X vs Wayland vs Mir). Don't attach too much value to it...
The LXDE and Razor-Qt projects merged and their future desktop environment releases are called LXQT and will migrate from using Gtk in their applications to Qt. For sure like on KDE you'll find that Linux distributions will favor to include Qt applications with it--but with LXDE/LXQT not even being in the top 10 of most used desktop environments and window managers, you should question anybody making a statement that this means anything for Gtk.
In any case, if you are a programmer look at both and decide for yourself which you like best. If you are a user, don't think too much on it--you can use applications written for either Gtk or Qt on your computer
There are many awesome applications written for both...