No, you will only get cartridges when your printer tells us you need them. HP Instant Ink cartridges are extra high capacity, and they contain months-worth of ink depending on your usage. You can print as much as you want as long as you are enrolled in HP Instant Ink.
I would suggest you return the depleted cartridges now. The printer will order cartridges as needed, the program generally tries to make sure you have about two months of ink on hand. Do you have replacement Instant Ink Cartridges on hand?
When the black cartridge is running low new ink should be ordered for you automatically. The new cartridge will include a new return packet for when the black cartridge runs out.