Early Quenya
patinka
noun. shoe, slipper
hyapa
noun. shoe
There are more words but they are from Tolkien's earlier conceptional periods. Tolkien likely changed these words as he evolved the aesthetics and completeness of the languages. You may even find languages that Tolkien later rejected. Do not mix words from different time periods unless you are familiar with the phonetic developments.
Below: Tolkien's Late Period (1950–1973). These are from Tolkien's late conceptual period — his most settled forms of the languages. Prefer these over the earlier ones above when composing new text.
A word appearing as ᴱQ. patinka “shoe, slipper” in the Qenya Lexicon of the 1910s under the early root ᴱ√PATA with derivatives having to do with rapping, tapping and clatter (QL/72).
Neo-Quenya: In Tolkien’s later writing √PAT seems to have meant “step” or “walk”, but I think ᴺQ. patinca can still be derived from this later sense, perhaps as a combination with the diminutive suffix ✶-inki (= “little-stepper”). However, I would use patinca only for “slipper”, and for “shoe” I would use ᴹQ. hyapat (Ety/SKYAP).