To capture the sql output messages (what you see in the “Messages” tab in SSMS) you can use this:
$results = (Invoke-Sqlcmd -ServerInstance "ServerName" -Database master -Query "BACKUP DATABASE [master] TO DISK = N'c:\master.bak' WITH Stats = 5" -Verbose | Out-Null) 4>&1 $results
Sql Server Powershell 