Welcome & congrats on your purchase!
If you have the auto dim & compass mirror, the compass should be on when the car's on. However, if there is a power button, press the power button. And if that doesn't work, then I dunno. I DO know that there are a couple mirrors out there that have auto dim, but no compass.
I bought a used autodim mirror from a salvage yard one time, that was off a cx-9. The mirror arrived and was the Gentex mirror, and looked exactly the same as what I've got now, but there was no compass in the upper right corner. I didn't know that was even a possibility.
Needless to say I took it back. Found a guy on here (mazdas247) that had one and sold me his since he was selling his cx-9, and it's the right one with the compass in the upper right corner.
Either you have the mirror with no compass, or maybe there's a wiring issue? You should have a green light on the bottom of the mirror while the car is turned on. When you press one of the homelink garage door buttons, the light turns red while you press the button.
Let me know.