The answer is Yes, if you are willing to open source your application.
According to the Qt-Project's own interpretation of the licenses they use, if you dynamically link to the Qt libraries your application can be either closed source or open. If you statically link, however, your application is subject to the terms of the LGPL.
The exact language the Qt Project uses is this:
In case of dynamic linking, it is possible, but not mandatory, to keep
application source code proprietary as long as it is “work that uses
the library” - typically achieved via dynamic linking of the library.
In case of static linking of the library, the application itself may
no longer be “work that uses the library” and thus become subject to
LGPL. It is recommended to either link dynamically, or provide the
application source code to the user under LGPL.
(http://qt-project.org/legal.html)
The suggestion in some of the other answers that the situation is "not at all clear" is simply untrue - The Qt Project has tried to be abundantly clear about in what circumstances usage of the LGPL license is permissible, and static linking is one of those as long as the app is also LGPL.
Since the original question specifies non-commercial (not necessarily open source), the asker will need to decide whether they can allow distribution under the LGPL (or GPL by extension, as the above page also says "LGPL can be converted to GNU General Public License").
