According to Capture Warning, Verbose, Debug and Host Output via alternate streams:
…if I wanted to capture verbose output in a script:
stop-process -n vd* -verbose 4>&1 > C:\Logs\StoppedProcesses.log
So, you would do something like
(Invoke-Sqlcmd -Query "PRINT 'Hello World!';" -ServerInstance $Server -verbose) 4> c:\temp\myoutput.txt
Where 4 is the “verbose” stream.