Yes. This should work. As you are aware, when you (or the Facebook contact) inputs their birthday into the contact, iOS creates a new calendar called "Birthdays". On both iOS and OSX you have control over the notifications that you receive for each birthday that is on that calendar.
Like you said you could can go and turn of the notifications which should prevent notifications from actually appearing. Note, however that the birthdays should still show in Notification Center.
You can also disable the Birthday calendar on your iOS device altogether by opening the Calendar.app, tapping "Calendars" in the upper left corner, and scrolling down to and tapping on "Birthdays". You will see the check mark next to it disappear and the birthdays will no longer show in your calendar.
It, of course, would be easy to turn them back on temporarily in the future if you wanted to look up someone's birthday.
You cannot silence the alerts of a single calendar in iOS. Alerts utilize Notification Center, and the Calendar options in Notification Center apply to all calendars. I know this is not the answer you are hoping for.
The best option, as mentioned in the comments above, is to disable the calendar on your work account, when not desired, and re-enable it again when you need it. To do this, go to Settings > Mail, Contacts, Calendars > Your Work Mail Account. Slide off Calendars. This isn't a good option, admittedly, as you will have to re-download your calendars every time it's slid back on.
Another option is to keep calendars turned off for this account, and use Exchange Outlook Web Access to access your calendars. This may be even a worse interface, however, as the OWA interface is not mobile-optimized.
My last suggestion is to use a third-party app to manage one of your calendars and the built-in Calendar app to manage the other. You can then disable Notifications on one of the two apps. You have to look potentially in two places for your calendar entries, but you will potentially get the granular control you need. Here are some calendar apps to examine: Calvetica, MiCal.
Best Answer
To the best of my knowledge you can't fix this. Hopefully Apple will modify this in an update at some point, but who knows. For now, the notification center is useless to anyone with a complicated calendar system.