Do I pay capital gains on the profit from the short-term sale of a house if the sale was necessitated by divorce?

Divorce (separation) is a specific event safe harbor per Section 1.121-3(e)(2)(iii)(D).

See https://www.law.cornell.edu/cfr/text/26/1.121-3

Enter the transaction in the Sale of Home section of the program. The Step-by-Step will walk you through the process.

IRS Publication 523 on page 6 describes an Unforeseeable Event such as divorce where you would be eligible for a partial exclusion due to a divorce - https://www.irs.gov/pub/irs-pdf/p523.pdf#page=6

Since you owned the home for 3 months prior to the sale your exclusion would be 3/24 of the $250,000 exclusion or $31,250.  If your gain was greater then you would owe capital gains tax on the amount over $31,250.

TurboTax will cover these types of situations when you enter the sale of the home.

Click on Federal Taxes (Personal using Home and Business)
Click on Wages and Income (Personal Income using Home and Business)
Click on I'll choose what I work on (if shown)
Scroll down to Less Common Income
On Sale of Home (gain or loss), click the start or update button

Or enter sale of home in the Search box located in the upper right of the program screen. Click on Jump to sale of home

You qualify for a partial exclusion, based on whichever time period is shortest:

1. How long you owned the home.

2. How long you lived in the home as your main home.

3. How long since you last used the exclusion.

Turbotax includes this calculation.  

See publication 523 for more information.

https://www.irs.gov/forms-pubs/about-publication-523

Any gain over the partial exclusion amount is a taxable capital gain, short-term in this case.