How many flip flops are required to build a digital circuit?
You do need four flip-flops to build the machine, but not simply because there are nine states. If you make a table that shows Q1 and Q2, then add as many bits as are necessary to give each state a unique representation, the result will be four total bits.

However, without some kind of reset input the machine could wake up in one of the seven (\$2^4\$ possible states minus the 9 that are defined) unused states. Your problem definition doesn't suggest how we should deal with that possibility.

I don't understand your "another argument". I don't know what you mean by "ports".