Bearing capacity on an inclined plane can be calculated by hand or in a simple spreadsheet calculation. I don't believe any specialty software is required.
There is a rational method in Bowle's 5th Edition textbook (1995) that I use (Chapter 4-9), theoretical. It was referenced in the 6th Ed. of AASHTO. The 4th Ed. of the textbook has the computer code if you wanted to write your own spreadsheet. Mr. Bowles was very generous.
AASHTO 8th Ed. provides another way to estimate, but no footing embedment (Leshinsky and Xie, 2016), theoretical. Supposedly an improved version of Meyerhof (1957). I haven't compared it to Bowle's method yet. Despite being in AASHTO, I haven't seen any validation.
Das's Shallow Foundations 2nd Ed. (Chapter 4-8) also provides a method (Saran, Sud, and Handa), theoretical.
Castelli and Motta (2009), limit equilibrium, theoretical.
Limit analysis and Soil Plasticity, 2007. Lower- and upper-bound solutions, theoretical.
For drained conditions, you can look at Shields, Chandler, & Garnier (1990). Centrifuge tests.
For undrained conditions and strip footings, you can look at Georgiadis, 2010. FEM.
The closest thing to software that's available is the spreadsheet by Wang and Skirrow, GeoEdmonton 2008. It uses Sokolovski's slipline solutions. Also theoretical. I was able to obtain the spreadsheet from the paper authors back in 2016.
Like others have said, this becomes a slope stability issue, so after you estimate a capacity, model the footing in/near the slope and verify stability at the estimated capacity (per limit state, where applicable).
We deal with this case often when widening roadway embankments where retaining walls will be constructed in existing fill. The sloping finish grade in front of the walls often range from anywhere between 1.5H:1V to 4H:1V.
