Emadelung it will be impossible to find a quad room in July for your budget which in euros is about 85 euros a night. You might,, big might.. find a one star somewhere far from central Paris.
You did not put where you are from on your profile.. so its a bit hard for me to figure out if you simply haven't stayed in hotels before.. or .. if you are perhaps from a country where 120 dollars seems like a lot of money for a room for four adults in a hugely popular large city like Paris.
I should explain that quad rooms are harder to find in Europe.. or at least in France.. then here in North America. Most hotel rooms are VERY small compared to what we are used to here.. they almost never will have king size beds, they often have double beds.. and their queen beds are sometimes two twins sheeted together.. Its VERY difficult to find a room with two queen, or even two double beds. And if you do they tend to get snapped up fast as many smaller budget hotels will only have one or two such rooms.
However I can suggest a few places for you to look into. They may be slightly over budget depending on your dates.
St Christophers Hostel
An apartment would not be a bad option , they can be economical when you consider that you will eat breakfast in and can make a few lunches for picnics.. but at this late date it may be hard to find one thats got good reviews ( I would never rent an apartment without good reviews)and most want a 7 day rental. But do look into them. I have only rented from Parisbestlodge but doubt they have anything left.. they are a small agency and super popular.. if you email and ask them you may hit it lucky though.. Still ,, its not going to be less then 100-150 EUROS a night for four people.
booking.com may be helpful to look at.. but always google hotel name and "reviews" and double check reviews from booking site to independent reviews.
Even hostels charge more then 25 euros a night in high season..