No, you won't get an entry in the SQL error log. A timeout is a client-side error, it's the client deciding that it's waited too long for a query to complete and telling SQL Server to abort the execution.
You'll need to identify what the application was running when it timed out, then investigating why that query is taking so long. Typical timeout periods are 30 seconds.
Gail Shaw
Microsoft Certified Master: SQL Server, MVP, M.Sc (Comp Sci)
SQL In The Wild: Discussions on DB performance with occasional diversions into recoverability