As a dog owner, I share your concern to protect your dog from exposure to anything harmful. However, no herbicide is lower in toxicity than Roundup is, as it is used. Roundup should not present a risk to your dog if you use it according to label directions. There are some vinegar and essential plant oil based herbicides available, but they are all more toxic than Roundup and will not affect the roots of weeds as they only act to "burn" exposed foliage. Flooding with large amounts of boiling water may penetrate enough to kill the roots of the weeds in your case, so that might be worth a try. Although you may still not wish to use Roundup, if you do, the best way to use it and minimize amount and exposure, would be to treat the freshly cut stems of the weed as the label directs. This would only require a very small amount of directed herbicide. No need to spray foliage this way.

I have lot of weeds in my backyard, is it safe to use Round-up weed killer? I have a dog.

Nilam

No matter which "weed" killer is used the directions will state that children and pets should be kept off the treated area for a minimum of 24 hours.
No synthetic poison is safe.