Yes but also no. The problem with fixed point arithmetic is a lack of dynamic range compared to floating point. Floats are great at representing both large numbers with limited precision and small numbers with high precision, but with fixed point you have to make a choice based on which kind of number you're trying to represent. Meaning you need to use a mixture of 8.24, 16.16 and 24.8 fixed point types (and appropriate conversions) depending on the context of the calculations that you're doing.

It's possible to write a game engine with that limitation, but there's no easy natural conversion from Quake's judicious use of floats to a fully fixed-point codebase. You'd have to redesign and rewrite the entire engine from scratch, basically.