No you cannot do the that i.e, you cannot enable logging for a specific user and leave the rest. You can check the activity from the Activity Monitor using the SQL Server Management Studio if you want to get the specific details of the user.
You can try this query to get the queries:
SELECT cr.DatabaseName
,s.session_id
,s.host_name
,s.program_name
,s.client_interface_name
,s.login_name
,s.login_time
,s.nt_domain
,s.nt_user_name
,c.client_net_address
,c.local_net_address
,cr.ObjName
,cr.Query
FROM sys.dm_exec_sessions AS s
INNER JOIN sys.dm_exec_connections AS c ON c.session_id = s.session_id
CROSS APPLY (
SELECT db_name(dbid) AS DatabaseName
,object_id(objectid) AS ObjName
,ISNULL((
SELECT TEXT AS [processing-instruction(definition)]
FROM sys.dm_exec_sql_text(c.most_recent_sql_handle)
FOR XML PATH('')
,TYPE
), '') AS Query
FROM sys.dm_exec_sql_text(c.most_recent_sql_handle)
) cr
where s.nt_user_name = '' -- filter here your user name
and s.session_id <> @@SPID
ORDER BY c.session_id
On a side note:
I usually use this query to get the list of queries executed in the past time
SELECT deqs.last_execution_time AS [Time], dest.TEXT AS [Query]
FROM sys.dm_exec_query_stats AS deqs
CROSS APPLY sys.dm_exec_sql_text(deqs.sql_handle) AS dest
ORDER BY deqs.last_execution_time DESC