SQL Monitor does enable traceflag 1204 yes. As a monitoring product, we have a relatively low impact on servers but that trace flag is essential so that we can detect deadlocks.
Regarding the T-SQL for the processes, it's possible from within SQL Monitor to enable a server-side trace. This is done via Configuration > Trace. After enabling this on the relevant server, future deadlock alerts should show trace information in the Performance Data section of the alert details page (in a tab named "SQL processes / Profiler trace"). Click the relevant SPID and trace should be displayed. Note that running the trace impacts the monitored server so it should ideally be disabled once you have seen enough information.
We have an enhancement request (SRP-4176) to move from traceflag 1204 to 1222. 1204 is required for SQL Server 2000 support but we are considering the option of using 1222 on more recent SQL Server versions.