On Windows you can only run Greengrass in Docker container.
But if the only thing you need is MQTT message exchange between your app and AWS IoT Core, you don't need Greengrass at all. I'd even say Greengreass would complicate things more.
Instead write MQTT client extension for your app and it will publish and subscribe directly to AWS IoT Core.
Here you have sample .Net MQTT publisher/subscriber application:
https://github.com/aws-samples/iot-dotnet-publisher-consumer
If you want to built in more functionality here is .NET SDK API reference:
https://docs.aws.amazon.com/sdkfornet/v3/apidocs/Index.html
Update:
On 12th November 2021 AWS released Greengrass version 2.5.0 which can be installed on Windows Server 2019 and Windows 10 devices.
Some Greengrass features may still not be supported though.
For current requirements and limitations refer to the documentation:
https://docs.aws.amazon.com/greengrass/v2/developerguide/setting-up.html#greengrass-v2-supported-platforms
Happy coding!