Nope. No foreseeable issues. You are able to fly intra-USA with no restriction (of course, have your passport on you for ID) - and there's no immigration intra-USA. You will just be dealing with the airline check-in counter if you have a bag to check, and then TSA (airport security) before you enter the departures area. Both will ask for your ID (passport, in your case)

Keep a copy of your return itinerary with you for immigration for when you land at SJC.

Enjoy your trip.