It sounds like ScheduleOnce would do the job for you - it is a paid application, (from $5/User/Month), with tight integration with Google calendar but there is a 30 day free trial.
To quote the web site:
Your personal booking page
Your booking page is your personal scheduling assistant and meetme.so/yourname is your personal booking page link. Add the booking page link to your email signature, print it on your business cards or simply send it to anyone who needs to schedule time with you. When people click your booking page link, they can easily see time slots in which you’re available but have no access to any calendar information – Not even to your free/busy view.
Quick and easy for your customers
You can use your Booking page with or without services, and accept bookings automatically or with your approval. Booking appointments with you only takes a few seconds. People select a time, provide their identification information, any additional information that you may have requested and submit.
Quick and easy for you
The appointment is automatically booked, or, if you would like to approve each appointment, you are notified to review the times and book it yourself. When booked, the appointment is automatically added to your calendar and confirmations are sent to your customers.
I have no personal involvement with the above product whatsoever and this recommendation is based solely on your stated requirements and the published specifications of the product.