No! There are compatibility problems with the current version of Apex (10.7).
For maximum compatibility, the Apex database is shipped in SQL Server 2000 compatibility mode. SQL Server 2012 does not support this mode.
A future version of Apex will be designed for new users that wish to install Apex on SQL Server 2012. In the meantime, SQL Server 2008 R2 works great and is the highest supported version.