Anything that ends up on the client in an unencrypted form can be accessed by the client. There's no way to avoid this. You have not control over what the client is doing with the responses that come from your server. The best you can do in this situation is obfuscate and/or try to make it more difficult for someone to access the key directly.
In situations like this, where you want to be able to allow your client to make direct calls to a third-party under your authority, you need some sort of token, ticket, or signing system which allows you to grant specific access to the client in a way that the 3rd-party provider can verify without giving the client your credentials.
I didn't spend much time looking into it and it's not really in the scope of this site to assist with the specifics of Googles API but as an example of how this works, it appears that this API provides a signature system. If I understand correctly, you would create a bare URL for the request that your client needs to make, and then using your credentials you get a signed URL from the API. This happens on your server, you would never provide your key to the client.
This signed URI would then be given back to the client and they can execute the call and the API will know that this is associated with your authority and billing. Presumably there is some sort of time limit on this signed URL. For those details, refer to the API documentation.