You gain sapphires when you reach certain levels on Momio. You can also gain 2 sapphires by saving your school info on Momio's Friend Finder.
Sometimes you can gain a sapphire by watching an advertisement on Momio. If you have the opportunity to earn a sapphire from watching an ad, it will be shown on the prize list before you begin watching ads. When you can no longer earn more sapphires, only diamond prizes appear on the list.
If you have Momio Plus, you will receive one sapphire every day for as long as you have Momio Plus. It's also possible to buy sapphires for real money.