$Code = "GOL"

This returns all the groups having "GOL" in their names:

Get-QADGroup "*$Code*"

This returns all the groups having _GOL_ in their names:

Get-QADGroup "*`_$Code`_*"

Both of these return nothing:

Get-ADGroup -Filter {Name -like '*$Code*'}

Get-ADGroup -Filter {(name -like "*$Code*")}