Would SQL Server Profile show triggers or other code which run as a side-effect of the query I'm profiling?
That should capture any ad hoc SQL or stored procedures any triggers associated with the table will fire, depending on the filters of your trace (I would be more concerned that it captures too much for you to sift through). 

But there is a much cleaner and less invasive way to know if a table has a trigger associated with it:

SELECT [Trigger] = tr.name
FROM sys.triggers AS tr
INNER JOIN sys.tables AS t
ON tr.parent_id = t.[object_id]
WHERE t.name = N'TableName';


There are multiple other reasons the profiler trace is unreliable - a trigger could be currently disabled, for example, or the trigger might only run further SQL statements if certain conditions are true.