The intended purpose of fertilizer is to increase soil fertility for plant growth. This question is too general to be answered easily but let's look at this.

Can fertilizer kill good insects? Yes, No and maybe. It is possible depending on the type of fertilizer, the formulation or the dose (concentration that fertilizer or any chemical whether conventional or organic can kill any insect whether good or bad. The dose can make a chemical a cure at a low level but a poison at a high level. Most fertilizers have nitrogen to make leaves green and phosphorous and potassium. Some are granular or they can be liquid or a slurry. If you apply a liquid directly on an insect you can kill it.
Questions to ask are what material?
what rate?
When to apply?
Do you need fertilizer? Have you done a soil test to determine if fertilizer is needed?
It should help to read the label of the fertilizer you intend to use and look for effects on beneficial insects. Follow label instructions and try not to apply when beneficials such as pollinators are present.