Question: Our .net application supports AD authentication, so our application will read from the GC connection string and credentials. We now have a client who has ADFS in their on-premise and wants to create a trust with our server which hosts the application. Do I need to write a separate code to setup as a claims-aware (or) can I use the same GC after creating the trust between client and our server?

Answer: 
you've answered your own question : "Our .net application supports AD authentication..." This is not the same s being claims aware, so yes you will need to modify the app. Once your app is claims aware they will create a relying party (that's your app) trust with some data that you will provide after it's been made claims aware. This trust will allow their server to send your app the token with whatever authorization data you require (and was defined when you set up your app). You should reach out to the clients infrastructure team to coordinate what data you would like and what they will provide.
You should ask this on <https://stackoverflow.com/>

If you do consider converting the app to be claims aware, I suggest you implement AD FS in your environment and create a trust between your AD FS and the client's ADFS. Your app will then needs to be converted to be claims aware and configure a trust between your app and your AD FS. You would be aiming to do <http://msdn.microsoft.com/en-gb/library/ee517273.aspx>

The domain trusts between two AD domains/forests are not the same as federation between two organizations using AD FS.

It looks like from your description your app is meant to work by looking up details in AD for users that exist in that AD. Your GC connection to your AD wont work unless that same AD has been populated with user accounts representing the client.

Start with the information here on converting your app to be claims aware using WIF <http://msdn.microsoft.com/en-gb/library/ee748475.aspx>

<http://www.cloudidentity.com/blog/> is a very good source on the latest guidance/information available on federated application development.