The base change of your quaternion algebra will be ramified at $\mathfrak{P}$ if and only the degree of the extension of completions $K_{\mathfrak{P}}/F_{\mathfrak{p}}$ has odd degree, i.e. case 3. A general way to see this kind of property is that the invariant in $\mathbb{Q}/\mathbb{Z}$ of the class in the Brauer group gets multiplied by the degree when you base-change the corresponding algebra. A more down-to-earth proof in the case of quaternion algebra is to use the standard presentation of the division quaternion algebra over the local field $F_{\mathfrak{p}}$ and to see that it contains all the quadratic extensions of $F_{\mathfrak{p}}$; in case 3 the local extension is trivial and so locally there is no base-change at all.

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**Edit:** Here are some references. Purely on $p$-adic fields, I guess Serre's *Local fields*. On quaternion algebras, there is the classic *Algèbre de quaternions* by Vignéras, but it is in French; there is the forthcoming book on quaternion algebras by John Voight, which is looking to be amazing; given your taste for geometry, I would also recommend the excellent *The arithmetic of hyperbolic three-manifolds* by Maclachlan and Reid, that contains very nice chapters on the local structure and the arithmetic of quaternion algebras. On more general central simple algebras and Brauer groups, I like the section in Milne's Class Field Theory notes, but this material is also contained in many textbooks.