Quartus II (not 2) was what older versions of the tool were called. If you take a newer project into one of those older versions, you'll need to recompile the design (same if you take an older project into a newer version of the software) in order to synthesize and run it in hardware.
For simulation, are you talking about using the old built-in simulation tool or a separate tool like ModelSim-Intel FPGA Edition? You really just need source design files and a testbench file to simulate.
So i actually don't need to synthesize my designs, it's just for simulations (educational purposes). My aim is to use Quartus Prime Lite edition, which is what I have available on my computer, to build my project and simulate it. However, since my instructor will be using Quartus II, I need to confirm they will be able to re-compile and simulate it.
And for the simulator, I'll be using ModelSim-Altera.