How can I use an adapter to get 120V from my generator to my house?

Now I understand your issue! Dumb of me, I apologize. You need 120 and you know those 4-prong plugs are associated with 240V somehow. You know 240 is not 120.

Actually the 4th wire makes it work. In North American residential wiring, a 4-prong outlet of these types (NEMA 14) actually contain both 120V and 240V. The 4th wire is a "midpoint" between 240V, giving you two separate "legs" of 120V each. 120V household circuits tap one or the other. The last pin is a safety ground.

Virtually all houses in North America are supplied this way, so this connection allows powering up your whole house. A 120V generator connection could only power half the circuits.

--

I'm guessing you inherited the generator interlock or transfer switch from the last owner. A 4-prong male is correct; that's called an inlet. Sounds like the last people did stuff right.

All you need is a cable to connect the generator's 4-prong outlet to the house's 4-prong inlet. Search both of them for a label like "NEMA 14-20" or "NEMA L14-30", or just compare the pins to an Internet search for those terms. From there, we can figure out which cable or adapters you need.