both "muck" and "meek" come from Latin "mucus"
preceded by Greek "myxa"

Merriam-Webster says different, Bill:

Main Entry: meek
Pronunciation: 'mEk
Function: adjective
Etymology: Middle English, of Scandinavian origin; akin to Old Norse mjukr gentle; akin to Welsh mwyth soft
Date: 13th century


And for muck:

Etymology: Middle English muk, perhaps from Old English -moc; akin to Old Norse myki dung


Having said which, there is a common "softness" here, and also a phlegmatic person could be called meek (in a slow and stolid way) - so maybe, just maybe.