The term Poisson integral formula may refer to any of the related formulas for harmonic (or holomorphic) functions on a disk (or in a ball, half space, etc) in terms of their boundary values. This is the formula that contains the famous Poisson kernel. When and where did Poisson publish his integral formula? I have a paper which says it was published in 1820 (without reference) and hours of searching has revealed nothing more.
I think that this is what you are asking for:
Mémoire sur la manière d'exprimer les fonctions par des séries de quantités périodiques, et sur l’usage de cette transformation dans la résolution de différents problèmes, Journal de l'École polytechnique, $18^e$ cahier, 11 (1820), p. 417–489.
Here you can find the scan of the original paper (in French, of course).
The so-called "noyau de Poisson" (Poisson kernel, clearly Poisson does not call it this way) appears at page 422.