Our foster child was born 10/3/2017 and placed with us 10/17/17, should we claim him on our 2017 taxes?

Yes, You can claim a foster child as a dependent as long as they are placed in your care by a placement agency, court order, or any government agency.  The child is treated as though the child lived all year as long as you can meet the all the test for a "qualifying child" during the period the child was born through the end of the year.

In your case as long as you were ordered by a government agency to care for the child you have passed the Relationship test.

Qualifying Child:

1. The child must be related to you. The child can be your son, daughter, stepchild, eligible foster child, brother, sister, half-brother, half-sister, stepbrother, stepsister, adopted child or an offspring of any of them.
2. Your child must be under age 19 or, if a full-time student, under age 24. There is no age limit if your child is permanently and totally disabled.
3. Your child must live with you for more than half the year, but several exceptions apply (ie. birth is considered living with you the whole year, away at school).
4. The child cannot provide more than half of his/her support.
5. You must be the only person claiming the child
6. The child must be a US citizen, resident alien, national, or resident of Mexico or Canada.
7. The child cannot file a joint return with his or her spouse. 
8. Click the image to enlarge and view for the IRS chart.