[2019] Does “Quick Play” Trigger the Critical Path/Story?

My question is this: Can quick play throw me into a mission beyond where I’m at, and can doing so throw off “My Story” in Tarsis by completing a critical path mission that I’m nowhere near?

Being an Xbox player, I’m trying to decide how to spend my 10 hours of EA Access next week. My goal is to meet some new friends and gather some experience and loot while doing it. With this in mind, I don’t want to outpace my friends on the story or get stuck doing free play the whole time. My thought for a compromise is to fill out other squads in quick play. Any help is appreciated!