To check quota usage via API, use the following experimental API.
Post a GET request to this URL: https://api.planet.com/auth/v1/experimental/public/my/subscriptions
In order to sign in, please insert your API key as username. Leave the password section blank.
The response lists details about all subscriptions associated with your account, including quota. You can use "quota_sqkm" and "quota_used" to calculate how much of your quota you have used.