The From name reflected on your email notifications sent to clients is pulled in from your YouCanBook.me booking page settings, not from your integrated Gmail account. You can modify the sending name in the Notifications section of your booking page.
Select the booking page you want to edit.
Go to Notifications > Actions, and for each email that sends to your booker (confirmation email, reschedule/cancel email, reminder email) you'll see the message "You've integrated Gmail. We'll send this email from (name) at (email)." Click on the name to edit the sending name on your notifications.