One of my PC's bought a hippogriff and was wondering: how long does it take to hatch? The text does not say anything about how long. How should I determine the hatching time frame or the incubation period? Can you even have a hippogriff as a companion?
You can definitely have a hippogriff as a companion--there's even an archetype for it.
But the text is silent on hippogriff egg incubation periods, so it's up to you as GM to decide.
If you're more of a storytelling GM, decide based on what's convenient timing for your game—otherwise known as "it incubates at the speed of plot."
If you're more of a simulation-minded GM, you still have to make it up, but you should find a number that you can accept as sensible. Perhaps research the incubation periods of very large and very small birds to find a ratio or power relation, then scale that up by your estimate of the mass of a hippogriff egg.
Either way, it's completely up to you—they're imaginary creatures, and the game doesn't care enough to say, so there is no higher authority than the GM on this question.
You could leave it to fate and ask your player to roll a dice for it. Roll 1D3 to determine days/weeks/month and then 1D20 for the actual period.
Alternatively, you could roll during the game: On crit the egg hatches. You may restrict it to no more than 3 or 1D4 attempt per game.