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The grass itself should not pose any threat to your dogs; however, the berries are mildly poisonous, more so with pups. Typically, upset stomach is most often experienced. You can remedy this simply by pulling the blooms or mowing to prevent the berries from forming.

Unfortunately, your only option here is to use a shovel and dig it up. However, if you want to get rid of the plant, most common weed killers will kill monkey grass.

First of all, the ripe, purple looking berries should be allowed to dry out thoroughly before taking out the seeds. Then they can be sown in early spring or fall.

If most of them are a broad leaf weed (like dandelions), you can use a broad leaf spectrum weed killer (it will be labeled as that).

If the weeds are grass like, they will need to be hand pulled as a grass specific weed killer will also kill the monkey grass.

Most common weed killers will kill monkey grass. Here is some information on both conventional and organic weed killers:
https://www.gardeningknowhow.com/plant-problems/weeds/conventional-weed-killers.htm
https://www.gardeningknowhow.com/special/organic/weed-it-and-reap-how-to-make-your-garden-more-appealing-without-chemicals.htm