None. Minster is from Latin monasterium. It and monastery are the same word, borrowed from Latin at earlier and later dates. A monastery was a place where a monk lived, Greek monachos; so ultimately it is from the Greek root mon-, "one, alone, single". But minister, now the most common word for "priest" in America, earlier meant "servant"; earlier, in Latin, it was one who is "lesser", as opposed to magister, "master, one who is greater"; thus its Latin root is min-, "small". Americans sometimes pronounce Westminster as Westminister. The confusion is understandable.

This is an example of linguistic convergence: words that are unrelated come to be similar both in sound and in meaning. I think it's an interesting subject (and you can see that mazy paths can lead from subject to subject, as well as from place to place).

Here is a small part of my personal collection:

Ra is an Egyptian sun-god: and a Polynesian sun-god.
English whole and Greek holos are not cognate.
French le, "the": Samoan le, "the".
Greek stylos, "pillar": Latin stilus, "stake".
Greek hydor, "water", and hidrôs, "sweat", are unconnected.
Greek duo, "two", and deuteros, "second", are from different roots.
Latin sus, "swine", and sumen, "sow, pig", are from different roots.
The name María is from Hebrew Maryam: but Mario is from Latin Marius.
The Persian word abrû means, but is not cognate with, "eyebrow" (and is not to be confused with âbrû, "honor").
Austria is the "eastern" land (German Oesterreich, "east kingdom"): Australia is the "southern" land (Latin auster, australis).