Yes, you DO need to install P-traps on ALL drains, including floor drains, even if you have to "bust concrete" to install the floor drains.
Sewage systems ALL generate nasty smelling gasses, including Hydrogen Sulfide [toxic], and others, and also EXPLOSIVE gasses [IF collect in adequate concentration].
Most local governmental agencies around the world have health codes[laws] which require the installation of P-traps. I can't imagine how your house got approved for occupancy without a plumbing inspection and final approval.
Without a whole lot of handyman skills, AND a whole lot of code studying, I recommend that you have this done by a reputable, licensed, professional Plumbing Contractor.
I suggest you get inspection and bids from 3 to 5 of your local licensed Professional Plumbers.