I am about to spray my lawn with weed killer. Does it matter if I put
this down first and then fertilize later?

You can do it in any order you like. One won't interfere with the
other. Unless your whole lawn is full of weeds, I'd use a tank sprayer
and just spot treat. If it is full of weeds, then I'd use a weed n
feed product and do both at the same time.