Powershell curl double quotes

Pipe the data into curl.exe, instead of trying to escape it.

$data = @{
    fields = @{
        project = @{
            key = "key"
        }
        summary = "summary"
        description = "description - here"
        type = @{
            name = "Task"
        }
    }
}

$data | ConvertTo-Json -Compress | curl.exe -X POST -u username:password -H "Content-Type: application/json" -d "@-"

curl.exe reads stdin if you use @- as your data parameter.

P.S.: I strongly suggest you use a proper data structure and ConvertTo-Json, as shown, instead of building the JSON string manually.

Easy way (for simple testing):

curl -X POST -H "Content-Type: application/json" -d '{ \"field\": \"value\"}'