Do you have a particular reason to fertilize? It typically is unnecessary unless you really like to mow grass.
If you really want to most general lawn fertilizers are not going to be harmful to dogs. They are mainly just various salts. Yes, if a dog ate several cups of fertilizer it may give them an upset stomach. Fertilizer with a pesticide, herbicide or insecticide, added could pose more of a health risk. But, still small. Apply just prior to rain or water in and risk to exposure is greatly diminished.
I use a liquid "weed b gon" by ortho to control broadleaf plants.
Did an all yard spraying years ago, now only have to spot control dandelions.
Once this is dry, it is safe for the dogs. Have been using this product for years, with no trouble for the dogs. I just make sure there is enough time for the product to dry, before I let the dogs on the yard.
John Deere landscapes has a product called "three way" for broadleaf. It is liquid 24-D that is very effective. As for fertilizer go with a lower nitrogen blend than what is typically offered by Scotts and others. When you hit your yard with high nitrogen it is more susceptible to urine burn spots. The combo of high nitrogen urine and high nitrogen fertilizer is too much even for irrigated lawns. Ask me how I know.
Talk to your Vet. I personally know many Vets( from all walks of life) that feel that 2,4,D is contributing to many of the Lymphomas in labs and goldens.
Like someone else said, I wait for time and a good rain and tread carefully if I know that anything has been sprayed or placed. I was shocked when I saw a guy spraying stuff at Bong in the dog training areas in all of the parking areas and not even putting up a sign. We went over and talked to him, and he said "Oh I will be careful around your vehicles." How am I supposed to get my dog back in the vehicle now? Who knows how many times I let me dog out without knowing that the place had just been sprayed. Why we can not have weeds in the Bong parking area is beyond me? Also, I was shocked when I found out, after the fact, that expansive areas were being sprayed in our Forest Preserve(Deer Grove) to kill one thing to try to get the prairie grasses to grow back. I know not to train my dog on the golf course, but now I have to be suspect of places that you think would be safe.
A spray application of a liquid to the weeds in the lawn is preferable to application to entire lawn with a dry fertilizer. You target the weeds only and reduce the total amount of pesticide you apply.
For dogs to get a dose large enough to make them sick they will need to come in contact with a lot of the product. This usually means they either drink the actual undiluted product or have it spilled on them.
The products of biggest concern are insecticides. They often act as nerve toxins. The biochemical pathways of insect and mammal nervous systems are very similar. The dose to kill an insect is much smaller than a dog or human. But, these are the products to watch out for.