I had the same problem on Windows when using the qsqlite.dll sqldrivers plugin. I was able to resolve the issue by using the following qt.conf:

[Paths]
Prefix = ./
Plugins = plugins

Also the folder structure was:

ExeDirectory\
ExeDirectory\plugins
ExeDirectory\plugins\sqldrivers

qsqlite.dll was located at:
ExeDirectory\plugins\sqldrivers\qsqlite.dll

qt.conf was located at:
ExeDirectory\qt.conf

Figured I would throw my two cents in.