Word for someone who is "quietly reliable"
Solution 1:
"steadfast".
This implies both reliability and a lack of drama. You could also go with "staunch", but that seems a little more likely to be not quiet.
Solution 2:
How about stalwart? As an adjective, it means "loyal, reliable, and hard-working." but you could also use it as a noun if you were specifying who the person was loyal to?